Note: I am providing an exhaustive answer as it is difficult to find this subject being answered to any degree of depth anywhere.
I think the answer can only be found by contrasting how these two covenants differ. These covenants largely differed: in their time in history, location, atmosphere, persons who inaugurated them, the persons who acted as mediator in them, subject matter, priests officiating under them, method of their dedication, success, permanency of their effects, aims, level of light in them, and the mode of governance that they established.
So the implicit question is, “How are they different in reality and how does that then specifically explain the difference in ‘feeling guilty’?"
The following comparisons show the difference between the Law and the Gospel, but it is not indented to imply there was no gospel in the Old Testament. The Old Testament does not just have the Laws given under Moses, it also had the Promise given to Eve (Genesis 3:15), renewed to Noah (Gen. 9:12-16), then made more clear to Abraham (Gen. 17), through David (Sam. 7:12-16), and continuing, running parallel to the Law but not joined in it by faith in the promise (Galatians 3:12), as the Law, as a covenant of works, pointed away from itself to the Promise. (Galatians 3:24)
Also the Prophets, under the Law, defended and applied the Law, while also directly pointed to the Messiah as that person who would fulfill all its demands for ever. (Acts 3:24) This is why Moses and Elijah both met Christ on the mountain during his transfiguration, whereby it was understood by them all that He would descend from that hill to the cross from which all the law and the prophets testified. (Luke 9:18-36) Moses was the representative of the Law, while Elijah was the representative of the Prophets as its most zealous defender.
In fact, the same contrasted comparison between Law and Gospel could be made with the Covenant of Promise made to Abraham, versus the Law given to Moses. If we lived before Christ, we could have written such an article. However, as the Promise in the Old Testament was pointing forward to the times of the Gospel, the comparison between Gospel and Law is more properly done with the perspective of the New Covenant being already established. There is much less benefit of making the comparison between the Law and the mere prefiguring shadows of the gospel.
Although the gospel is in the Old Testament, running parallel to the Law, and directed sinners away from itself by commanded full obedience (Leviticus 18:5, James 2:10, Mathew 19:21) it is a different subject not under the focus of the task at hand. We only desire that it should be clear from the outset that the kernel of the gospel was always in the Old Testament as the Covenant of Promise as distinct from the Law.
It is through the Covenant of Promise that men could always be saved even before the Law was introduced (Galatians 3:17). Yet it must also be recognized that even those who were saved by faith, apart from the works of the Law, while being made subject to the Laws of Moses at that time, could not have fully enjoyed the ‘liberty of the gospel’ as will be shown under this comparison. If they could have, there would have been no reason to change anything. (Hebrews 8:7) We would have kept the daily sacrifice for sins ‘looking back at Christ’ who they prefigured, just as an ancient man, full of faith, may have presented his beast ‘looking forward’ to the Messiah. The truth is that many under the Old Covenant were not circumcised of heart, but flesh only. It was to this sort of person that the Law was principally made for and who it principally speaks to (Hebrews 10:1-3):
We also know that law is made not for the righteous but for lawbreakers and rebels, the ungodly and sinful, the unholy and irreligious; for those who kill their fathers or mothers, for murderers (1 Timothy 1:9)
Under the same token we must add that although the Law was removed in Christ, as to its outward form and power over the conscience, its ruling jurisdiction has always existed since the fall, for it merely took the law of conscience that condemns sinners by nature and expounded on it under commands. (Romans 1:18-21) Therefore, the Law has always been in place since the fall and will always be in place as long as sinners remain. We are all born in sin, under law, even if Moses had not manifested what those Laws were, and all who believe in Messiah, since the fall of man to the return of Christ, are freed from that Law inwardly and are transplanted into grace by the gospel. (Colossians 1:13, Romans 4:1-3)
Those saints of Old who had to suffer under the legal, ceremonial and civic Laws of Moses, being somewhat contrary to their inward liberty and circumcision of heart, did so as a theatrical service for those who had not yet believed. As Christ had not yet come, the Law assumes all have not yet believed until He came. For although both sinner and saint were circumcised as a child of Moses, the law had to assume they were one, or the other. It is clear that it assume a state before Christ was raised. In a similar but less strict manner, even in the New Covenant believers assume the salvation of those who claim faith and allow them to take part in the sacraments of Baptism and the Lord’s Supper. Even a Devil we be assumed a saint under the gospel, if faith in Christ is claimed without any scandalous sins to question the claim. This explains why a Jew under the Old Covenant would die if attempting to draw near to God in the Holy of Holies. It was assume that that Messiah had not yet come to tear that curtain from top to bottom (Mathew 27:51, Leviticus 16:2).
How the Covenants Differ in Terms of the Spirit of Guilt:
1-They differ in time:
The Old Covenant refers to the covenant inaugurated during the time of Moses. It was a covenant of Law that many have defined under the categories: moral, civil and ceremonial. When the words ‘The Law’ are used in scripture it most commonly refers the first five books of the Bible with respect to the Laws of Moses. The 'moral law' is generally tied to the Ten Commandments, the 'civil law' to various capital punishments and rules related to the 'eye for an eye' precept of justice. The 'ceremonial law' has to do with the Levitical Priesthood and the entire God ordained rules of worship that were conducted in God's Temple.
The New Covenant was delivered in the “in the latter days,” (Hebrews 1:1-2) and “in the dispensation of the fullness of times.” (Ephesians 1:10) and it formally made the Old Covenant obsolete during the actual death of Christ when the inner certain of the Temple was torn from top to bottom (Mathew 27:51).
2-They differ in the location that they were inaugurated:
The first was declared on Mount Sinai. This mountain is Arabia was the home of Ishmael and his race. The Arabian name for Mount Sinai was Hagar. The New Testament was declared on mount Zion, and the law of it went forth from Jerusalem (Isaiah 2:3).
The Apostle Paul takes special notice of this difference of location, because ‘Hagar’ was also the name of the slave who gave birth to Ishmael, Abraham's first lawful child. Hagar's child was a child of Abraham by law, but Isaac was the child that God ‘promised’ to Abraham and his wife in their very old age. Therefore, the Old Covenant is associated with slavery and bondage under the name Hagar the slave. The New Covenant, or the Covenant of Promise, originally established with Abraham, but not fulfilled until Messiah appeared in the flesh, is associated with ‘freedom’ and ‘liberty’. It was promulgated from Jerusalem, not from Hagar.
22 For it is written that Abraham had two sons, one by the slave woman and the other by the free woman. 23 His son by the slave woman was born according to the flesh, but his son by the free woman was born as the result of a divine promise.
24 These things are being taken figuratively: The women represent two covenants. One covenant is from Mount Sinai and bears children who are to be slaves: This is Hagar. 25 Now Hagar stands for Mount Sinai in Arabia and corresponds to the present city of Jerusalem, because she is in slavery with her children. 26 But the Jerusalem that is above is free, and she is our mother. 27 (Galatians 4:22-27)
3-They differ in the atmosphere of their establishment.
The Law of Moses was delivered on:
a mountain that can be touched and that is burning with fire; to darkness, gloom and storm; 19 to a trumpet blast or to such a voice speaking words that those who heard it begged that no further word be spoken to them, 20 because they could not bear what was commanded: “If even an animal touches the mountain, it must be stoned to death.”[c] 21 The sight was so terrifying that Moses said, “I am trembling with fear.” (Hebrews 12: 18-21, See also Deuteronomy 5:23-27)
By this means a spirit of fear and bondage was administered to all the people, so as that they chose to keep at a distance, and not draw near to God, (Deuteronomy 5:23-27).
The message from Jerusalem has a free spirit of pure worship and praise without any gloom whatsoever:
22 But you have come to Mount Zion, to the city of the living God, the heavenly Jerusalem. You have come to thousands upon thousands of angels in joyful assembly, 23 to the church of the firstborn, whose names are written in heaven. ((Hebrews 12: 2-23)
4-They differed in the persons that declared them and promised there with:
In the Old Covenant the Law was given to Moses by angels and the whole church became subject to the authoritative rule of angels under it. (Acts 7:53, Galatians 3:19)
In the New Testament we find this subjection to the angels was not meant to last. (Hebrews 2:5) Rather the gospel was declared directly from the Son of God.
14 The Word became flesh and made his dwelling among us. We have seen his glory, the glory of the one and only Son, who came from the Father, full of grace and truth. (John 1:14)
17 For the law was given through Moses; grace and truth came through Jesus Christ. 18 No one has ever seen God, but the one and only Son, who is himself God and is in closest relationship with the Father, has made him known. (John 1:17-18)
5-They differ in their mediators.
The mediator of the first covenant was Moses (Exodus 32:11-14, Hebrews 3:5) Moses stood between the people and God because the people could not endure the frightening terror and horrible gloom of His presence.
24 And you said, “The Lord our God has shown us his glory and his majesty, and we have heard his voice from the fire. Today we have seen that a person can live even if God speaks with them. 25 But now, why should we die? This great fire will consume us, and we will die if we hear the voice of the Lord our God any longer. 26 For what mortal has ever heard the voice of the living God speaking out of fire, as we have, and survived? 27 Go near and listen to all that the Lord our God says. Then tell us whatever the Lord our God tells you. We will listen and obey.” (Deuteronomy 5:24-27)
As Moses was a mediator it is sufficiently clear the Law treated those who were under it a sinners who must not approach God.
The mediator of the New Covenant is the Son of God himself. For “there is one God, and one mediator between God and
5 For there is one God and one mediator between God and mankind, the man Christ Jesus (1 Timothy 2:5).
6-They differ in their subject matter.
The Old Covenant renewed the commands of the covenant of works. The Law threatened death for sin, any kind of sin with eternal death. The Law promised life only to those who fully obeyed it with sinless obedience.
The Ten Commandments is the very words of the covenant (The Covenant of Promise to Abraham is not included under the Law, as it was actually the gospel in its pre-Christ form):
28 Moses was there with the Lord forty days and forty nights without eating bread or drinking water. And he wrote on the tablets the words of the covenant—the Ten Commandments. (Exodus 34:28)
5 Keep my decrees and laws, for the person who obeys them will live by them. I am the Lord. (Leviticus 18:5)
10 For all who rely on the works of the law are under a curse, as it is written: “Cursed is everyone who does not continue to do everything written in the Book of the Law.” 11 Clearly no one who relies on the law is justified before God, because “the righteous will live by faith.” 12 The law is not based on faith; on the contrary, it says, “The person who does these things will live by them.” 13 Christ redeemed us from the curse of the law by becoming a curse for us, for it is written: “Cursed is everyone who is hung on a pole.” 14 He redeemed us in order that the blessing given to Abraham might come to the Gentiles through Christ Jesus, so that by faith we might receive the promise of the Spirit. (Galatians 3:10-14)
Therefore under the strict Law had no promise of grace, to communicate spiritual strength, or to assist in obedience. It was therefore hopeless under the Law to obtain eternal life, so in its stead material blessing in the land of Canaan became inseparable with it.
7-They differ in the manner of their dedication.
It is the solemnity and manner of the confirmation, dedication, and sanction of any promise or agreement, that gives it the formal nature of a covenant or testament.
The Old Covenant was sanctioned by the sacrifice of ‘beasts’, whose blood was sprinkled on all the people. (Exodus 24:5-8).
The New Testament was sanctioned by the sacrifice and blood of Christ himself, which became ‘the blood of the covenant’, which we remember at the Lord’s Supper.
8-They differ in the priests that were to officiate before God in the behalf of the people.
In the Old Covenant, Aaron and his posterity alone were to discharge that office. On account that those old priests were sinners and had to make sacrifice for their own sins, the priest were many.
In the New Covenant, the Son of God himself is the only priest of the church. Not only does his priesthood go uninterrupted so that He is a Priest forever, but as a Priest without sin He can officiate those things that pertain to our spirit, making us spiritually clean:
22 let us draw near to God with a sincere heart and with the full assurance that faith brings, having our hearts sprinkled to cleanse us from a guilty conscience and having our bodies washed with pure water.(Hebrew 10:22)
This directly speaks to the ‘guilty feeling’ that we will return to in the conclusion.
9-They differ in the sacrifices whereon the peace and reconciliation with God was proposed and was depended on.
In the Old Testament, the sacrifices offered were merely animals. They were designed to prefigure the Messiah as the only proper sacrifice for sin. They could only therefore purify the worshiper externally in order that worshippers may continue to enter into the Temple after they had encountered unclean things, meats, mold, etc.
In the gospel the Son of God offered His own Holy soul and body for the sins of the world so that anyone who believed in Him might have eternal life. (John 3:16)
10-They differ in their permanency and effectiveness of their writing.
All covenants were of old solemnly written in tables of brass or stone, where they might be faithfully preserved for the use of the parties concerned.
The principal part of the law was “carved in tables of stone” which were kept in the ark (Exodus 31:18, Deuteronomy 9:10; 2 Corinthians 3:7). However God ordained its foundation to be destroyed as soon as it was made, in order to show that none could meet its requirements and that never was it to be viewed as permanent.
In the New Covenant the words were not written on stone, rather they were written in the heart. This largely distinguished the two covenants and shows that the New Covenant is not just an unfolding of the same covenant, with the same nature. Rather the Old Covenant is something God himself had ‘fault’ with and God himself made obsolete.
7 For if there had been nothing wrong with that first covenant, no place would have been sought for another. 8 But God found fault with the people and said:
“The days are coming, declares the Lord, when I will make a New Covenant with the people of Israel and with the people of Judah. 9 It will not be like the covenant I made with their ancestors when I took them by the hand to lead them out of Egypt, because they did not remain faithful to my covenant, and I turned away from them, declares the Lord.
10 This is the covenant I will establish with the people of Israel after that time, declares the Lord. I will put my laws in their minds and write them on their hearts. I will be their God, and they will be my people. 11 No longer will they teach their neighbor, or say to one another, ‘Know the Lord,’ because they will all know me, from the least of them to the greatest. 12 For I will forgive their wickedness and will remember their sins no more. ”
13 By calling this covenant “new,” he has made the first one obsolete; and what is obsolete and outdated will soon disappear. (Hebrews 8:7-13, Jeremiah 31:31-34)
11-They differ in their aims.
The principal end of the Law was to discover sin, increase it, to condemn it, and to set external bounds to it through fear of punishment. Therefore it was always accompanied with judgments and punishments.
20 The law was brought in so that the trespass might increase. (Romans 5:20)
Fear prevented sinners from sinning, but yet increased the desire making them ‘locked up’ and leading to Christ from God’s wrath:
23 Before the coming of this faith, we were held in custody under the law, locked up until the faith that was to come would be revealed. (Galatians 3:23)
As the Law was employed in this manner as to hold sinner under the torment of their sin, the scripture refers to the Law as a Law of sin and death. For although the Law was Holy and good, by the sinfulness of flesh it became a harsh lord over the damned doing nothing else but driving to despair in order to smash the self-righteousness of men who suppress the truth.
The aim of the New Covenant is to declare the love, grace, and mercy of God and by doing so give: repentance, remission of sin, and eternal life. (John 3:6)
In contrast to the law of sin and death, the gospel is a law of the Spirit of life. As the law brought death on mankind by one man, and so death reigned unto all, the righteousness of Christ and the obedience of one man, brings righteousness to all who believe. Yet the power of this new law is sin in that it overwhelmed manifold sins, deaths, hells, Satan’s and curses. It came after many sins had increased the aggravation of the first sin. Yet by One man, all these offences, death, hell, curses, etc were killed by Christ death. Therefore the death of death, the death of sin, the death of Satan, the death of guilt, the death of mortality were all made by the death on the Messiah.
How was this done? All sin was hammered into the flesh and soul of Christ making Christ the biggest sinner. And as a man who submitted and perform everything the Laws of Moses required including the submission to the baptism of John and all additional requirement that the Law had upon the Messiah as the literal 'beast' to truly remove sin under its legal bench, Christ not only took upon our sin, but released up his perfect obedience that was in turn charged onto our account.
Now if our state under Law is to be found with Christ’s very own righteousness than the law has no more purpose, abandoning its charge of us.
The Law can't accuse those who have been made perfect. The Law can only see sinners in Christ as having obeyed all its precepts from cradle to grave.
Therefore if the Devil tries to resurrect the Law and pull Christ down from heaven pointing His finger our way and saying, "Yes but you have committed many big sins, and someone must pay!". The believer san say to the Law and the Devil who has tried to resurrect it, "Yes, and what are you going to do about it!". The devil can only than flee with his attempts and seek someone else to devour.
12-They differed in their effects.
The first covenant was a “ministration of death” and “condemnation,” it brought the minds and spirits of those that were under it into servitude and bondage. This slavery and bondage was typified under the practices of the temple where none could approach God. The Most Holy Place in the temple, where God’s presence was, was accessible only to the high priest and only once a year. Approaching God in any other way than that prescribed by God was forbidden.
2 The Lord said to Moses: “Tell your brother Aaron that he is not to come whenever he chooses into the Most Holy Place behind the curtain in front of the atonement cover on the ark, or else he will die. (Leviticus 16:2)
Those under the Laws of Moses are considered comparatively as and underage sons, or even slaves, for even the servants of a home can boss around children in the household:
4 What I am saying is that as long as an heir is underage, he is no different from a slave, although he owns the whole estate. 2 The heir is subject to guardians and trustees until the time set by his father. 3 So also, when we were underage, we were in slavery under the elemental spiritual forces of the world. 4 But when the set time had fully come, God sent his Son, born of a woman, born under the law, 5 to redeem those under the law, that we might receive adoption to sonship. 6 Because you are his sons, God sent the Spirit of his Son into our hearts, the Spirit who calls out, “Abba, Father.” 7 So you are no longer a slave, but God’s child; and since you are his child, God has made you also an heir. (Galatians 4:1-7)
As the Law (though perfect and holy) increased fear of death due to its condemnation the Devil himself used the Law to increase sin and bring them in further subjection to his malice:
14 Since the children have flesh and blood, he too shared in their humanity so that by his death he might break the power of him who holds the power of death—that is, the devil — 15 and free those who all their lives were held in slavery by their fear of death. (Hebrews 2:14-15)
So it was truly the “letter that killed” them.
In the New Covenant spiritual liberty is the immediate effect. This contrast between covenants is often attributed to the gift of the Spirit, which is a Spirit of liberty in contrast to bandage and fear.
15 The Spirit you received does not make you slaves, so that you live in fear again; rather, the Spirit you received brought about your adoption to sonship. And by him we cry, “Abba, Father.” (Romans 8:15)
17 Now the Lord is the Spirit, and where the Spirit of the Lord is, there is freedom. (2 Corinthians 3:17)
13-The level of light to relieve the conscience is different.
In the Old Covenant they did have the Promises of Abraham which was the gospel and no law. Some, and at time many were able to enter into the rest of New Covenant from afar. Even in those days faith in the promised Messiah was just as effective in providing eternal life as is faith looking back at Messiah. However even the Prophets who predicted his coming were under darkness about what they spoke.
17 For truly I tell you, many prophets and righteous people longed to see what you see but did not see it, and to hear what you hear but did not hear it. (Mathew 13:17)
Even John the Baptist who was the greatest of those prophets since He actually could see He who all the Prophets spoke of was considered less than the least of those who believed after under the New Covenant:
11 Truly I tell you, among those born of women there has not risen anyone greater than John the Baptist; yet whoever is least in the kingdom of heaven is greater than he. (Mathew 11:11)
In the New Covenant believers are able to obtain much clearer knowledge and faith of His incarnation, sufferings, and sacrifice, through which he made atonement for sin. This gives them liberty and boldness in their obedience:
16 Let us then approach God’s throne of grace with confidence, so that we may receive mercy and find grace to help us in our time of need. (Hebrews 4:16)
The great difference of light and glory in the two covenants renders the Old Covenant without glory by comparison:
10 For what was glorious has no glory now in comparison with the surpassing glory. (2 Corinthians 3:10)
Although there was some grace and truth in the Old Covenant, for all the ceremonies prefigured Messiah and the Law is perfect and holy, yet by comparison the scripture speaks as there was no truth or grace before Jesus Christ:
17 For the law was given through Moses; grace and truth came through Jesus Christ.
14-The mode of governance in the level external details and rites are different:
In the Old Covenant many ceremonies were required because each was not good enough to foreshadow Messiah. Therefore many laws were added and increased by the yoke of a multitude of laws, rites, and ceremonies, imposed on them. This made the whole of their worship a burden to them, and insupportable:
10 Now then, why do you try to test God by putting on the necks of Gentiles a yoke that neither we nor our ancestors have been able to bear? (Acts 15:10)
This burden, in addition to sin itself, must have been at least in part the simple appeal of Christ’s call:
28 “Come to me, all you who are weary and burdened, and I will give you rest. 29 Take my yoke upon you and learn from me, for I am gentle and humble in heart, and you will find rest for your souls. 30 For my yoke is easy and my burden is light.” (Mathew 11:28-30)
In summary when we understand that the Law was meant to lead to Christ it must now be obvious that it directed away from itself to the Promise made to Abraham.
He suffered them to all sort of fears, bondage, guilt and restlessness so that they might not rest in that state, but continually look out after deliverance. Even those who entered into that faith, they saw things too faintly to full obtain the same freedom and boldness which is now proclaimed under the gospel.
Under the New Covenant there is freedom from the commanding power of the law as far as it required perfect obedience and increased our sin by making us aware of sin without providing any help to obey. On this account we can consider all our good deed under an accusing conscience as ‘garbage’ and in full liberty cling to Christ’s righteousness alone apart from any of our own efforts of faithfulness and holiness:
8 What is more, I consider everything a loss because of the surpassing worth of knowing Christ Jesus my Lord, for whose sake I have lost all things. I consider them garbage, that I may gain Christ 9 and be found in him, not having a righteousness of my own that comes from the law, but that which is through faith in Christ—the righteousness that comes from God on the basis of faith. (Philippians 3:8-9)
28 For we maintain that a person is justified by faith apart from the works of the law. (Romans 3:28)
We are free from all the condemnation itself because Christ absorbed all its threats and curses:
From surveying the difference between the Law and the Gospel we have more than enough knowledge to accurately answer why 'guilty feelings' are encouraged under the Law as part of its bondage and the removal of those feelings by the 'sprinkling of blood' as one of the rests tendered in the gospel.
Since the gospel frees the conscience from the power, condemnation, curse, fear of death, ceremonial reminder of guilt, terrifying civil of punishments and any other thing that might make us feel unworthy of joyfully approaching the throne of grace with boldness in a spirit of liberty we have no reason to carry on under the gloomy dread of guilt.
But someone may ask, "Yes but if you sin you must feel guilty?" But this question is not reasonable. Yes we sin every day yet we do not feel guilty every day as Christ was punished for the sin. "Oh No!", some might say. "This is encouraging sin that grace may abound!". But not true again when looking at the scriptures. A liberated conscience from the curse of the Law brings true holiness and guilt merely increases sin, even when its outward form is bound in prison under fear. Therefore the person who claims these things only deceives himself and increases his own sin, while judging those who are free. This is why Hagar and Jerusalem are symbols of people today.
The real relationship of sin to guilt, is that when we do not believe in God's grace with sufficient faith as to thoroughly cleanse the conscience then our minds become subject more and more to the flesh which is still under the Law. As our mind become more and more fleshly we can't get rid of guilt no matter what we do, for guilt can't be washed by the blood of beats which include any fleshly thought we may use to comfort ourselves. No, only faith can clean the conscience from the guilt of sin, and once the conscience is cleansed and a joyful praise resounds in the liberated soul, then and only then does faith go on to produce to holiness and works.
Guilt is not cleansed by works, but by faith. This is the practical application of living by the Spirit and not by the works of the Law.
Every other notion is madness and sin.
And by all these and any other instances of spiritual liberty, the gospel frees believers from that “spirit of slavery", "guilt" and "fear” administered under the Old Covenant.