Part A - Why do non-LDS Christians accept the Testimony of the Apostles?
Contrast, Not Comparison It must first be considered that in this discussion we can not compare the two groups of witnesses, we can only contrast them! The differences are so great, that they are two different classes of people!
1. Appointed by Jesus First, as Xartoon noted, after a night of prayer, Jesus Himself appointed the Twelve Apostles (Matthew 10). Not an angel, nor a High Priest, but Jesus. This gives credibility from the beginning. Even Judas was picked for the purpose of fulfilling what the Apostles witnessed: the Death and Resurrection.
2. Content of Message Notice the content of the message of the Apostles: what they testified to was open and public events; nothing secret about it all! They were not relating what was done in a cave (like Mohammed), nor what was done in a secluded forest at Cumorah (like J. Smith).
They were testifying to events in the life of Jesus that the whole country of Judea was buzzing about. Open demonstration of miracles, public healings, witnessed resurrections from the dead, multiplication of food for whole crowds!
So the Apostles testimony was easily confirmed by multitudes of people. What the Apostles testified to could not be denied at all. The content of their message was known throughout the region.
Jesus answered him (high priest), "I spoke openly to the world; I ever taught in the synagogue and the Temple where the Jews always resort, and in secret I have said nothing. (John 18:20; Luke 21:37)
The king (Festus) knows these things before whom I also speak freely: for I am persuaded that none of these things are hidden from him; for this thing was not done in a corner. (Acts 26:26)
3. Totality of Testimony Note also that the Apostles were not just accident witness of just one tragic event that they were trying to recall. Each Apostle had to have been a disciple of Jesus for 3 1/2 years! (Acts 1:21-22) They had to have been witnesses over and over again of the miraculous ministry of Jesus leading up to the Resurrection...which they also witnessed with hands on empirical verification. (Luke 24:36-43, 1 John 1:1-3)
What they testified to was not secret meetings with an angel or dead saints, on one or more occasions, but a full number of years of testimony on and on. Day after day month after month, year after year.
4. Testimony of Enemies What the Apostles testified to was known by Christ's enemies, as well. They couldn't deny it, as the Apostles pointed out. The Apostles' testimony was borne out by the rulers who hated Him. It was therefore solid testimony. Notice what Peter said to the Roman centurion:
The word you know, which was published throughout all Judea, and begun from Galilee, after the Baptism of John; how God anointed Jesus of Nazareth with the Holy Ghost and with power, Who went about doing good, and healing all that were oppressed by the devil, for God was with Him.
And we are witness of all things which He did both in the land of the Jews, and in Jerusalem; whom they slew and hanged on a tree. Him God raised up and showed Him openly, not to all the people, but unto witnesses chosen before by God, even to us who did eat and drink with Him after He rose from the dead! (Acts 10:37-41)
And when Herod saw Jesus, he was exceeding glad; for he was desirous to see him of a long time, because he had heard many things about Him...(Luke 23:8)
And when Felix heard these things, having perfect knowledge of the Way, he deferred them... (Acts 24:22)
Paul said, I am not mad, most noble Festus, but speak forth the words of truth and soberness. For the king knows of these things before whom I speak freely. For I am persuaded that none of these things are hidden from him, for this thing was not done in a corner. (Acts 26:25-26)
5. Testimony Confirmed by Hundreds It is recorded that the testimony of the Twelve was confirmed by hundreds of other people. We don't just have their say so, but we have objective confirmation that makes it reasonable to accept these men's testimony. ("delivered and received" was a rabbinical saying)
For I delivered unto you first of all that which I also received...He was seen by Cephas, then by the Twelve, after that He was seen by above five hundred brothers at once, of whom the greater part remain unto this day...After that He was seen by James, then by all the of the Apostles... (1 Corinthians 15:3-7)
There were also multitudes of people all over the land of Judea, Perea, Samaria, and Galilee who would readily confirm the testimony of the Apostles concerning the miraculous power of Jesus. Jesus had healed people in almost every village He went to.
6. Cross-Examined Next, recall that these testimonies of the Apostles were cross-examined in judiciary and regal halls of government. (Acts 5:27, 12:1) The Apostles were given the third degree; and it wasn't like the modern pristine courtrooms with civil rights attorneys protecting them! Rather the Apostles were scourged, beaten and whipped to see if they would change their testimony. The Christian witnesses had stripes on their back, yet continued to present the Good News of Jesus. Some were jailed.(Acts 4:3)
...when they had called the Apostles, and beaten them, they commanded that they should not speak in the Name of Jesus, and let them go. (Acts 5:40)
And they cast (Stephen) out of the city, and stoned him. (Acts 7:58)
At that time there was a great persecution against the Church which was at Jerusalem, and they were scattered abroad...(Acts 8:1)
7. Prophetic Witness Another reason we know that the Apostle's witness is reliable is the confirmation of the Old Testament prophets. What the testimony of the Apostles' was, lined up with what was predicted hundreds of years previous!
For David speaks concerning Him, I foresaw the Lord always before my face...thou wilt not leave my soul in hell, neither will you suffer thine Holy One to see corruption (i.e. the resurrection) (Acts 2:25-27)
Christ, whom God raised from the dead, even by Him doth this man stand here beore you whole: This is the stone which was set at nought by you builders, which has become the headstone! (Acts 4:10-11; Psalms 118:22)
To Him all the prophets gave witness, that through His Name whosoever believes in Him shall receive remission of sins. (Acts 10:43)
We have a more sure word of prophecy; whereunto you do well that you take heed...(2 Peter 1:19)
8. Miraculous Confirmation Another reason that Christians put their confidence in the testimony of the Apostles is the working of miracles to confirm their teaching and evangelism. Jesus said this would happen, and the ministry of the Apostles bear this out.
Go into all the world and preach the Gospel to every creature...these signs shall follow them that believe... And they went forth, and preached everywhere, the Lord working with them, and confirming the Word with signs following. Amen! (Mark 16:15-20)
So when they had further threatened them, they let them go...for all men glorified God for that which was done. For the man was above forty years old on whom this miracle of healing was done. (Acts 4:21-22)
And believers were the more added to the Lord, multitudes both of men and women. Insomuch that they brought forth the sick into the streets, and laid them on beds and couches that at the least shadow of Peter passing might overshadow them. There came also a multitude out of the cities round about unto Jerusalem, bringing the sick folks, and them who were vexed with unclean spirits, and they were healed every one! (Acts 5:14-16; see also Acts 8:4-8, 9:33-35, 10:44)
What of any of the world religions or sects ever had witnesses who gave sight to the blind, made the lame to walk, or raised the dead to life? But these miracles of Christianity are unique to the Christian religion, confirming the veracity, credibility, and authenticity of the witness to the Gospel of Christ and the Apostles. (Hebrews 2:3-4)
9. At the Place of Events Finally, note that most of this wonderful testimony of the witnessing Apostles was performed, taught, declared, right at the location where it happened: at Jerusalem.
If men were to try to commit fraud they would have taught the story in some far way land where no one could deny it occurred! Where there would be no witnesses to the contrary. Once upon a time in a far away land, there was a... But NO! The Apostles stayed for a long while in Jerusalem (until the destruction of the Temple)! And they even were boldly preaching in the Temple, right under the noses of the High Priest and rabbis, and Roman rulers. Those men could not deny that Jesus was crucified, and that the tomb was now empty...and that miracles were being done in His Name.
Part B - Why do non-LDS Christians reject the Testimony of the eleven Mormon witnesses?
Many Reasons Reasons why the Mormon witness are not accepted as reliable, and must be rejected, are many! A contrast with the Early Church Apostles demonstrates this clearly. Consider:
1. Public Revelation As was adequately mentioned by another answer, Christian theologians (both Catholic, Protestant, and Orthodox) realize that "Public Revelation" ended with the death of the last Apostles of Jesus (2 thousand years ago!). Mormon revelation falls far out of that time frame. Witnesses to any alleged revelation in the 1830s is therefore to be rejected.
2. Jesus Appointed The N.T. Apostles were clearly appointed by Jesus, after a night of prayer to the Father...and they were the ones who became the Christian witnesses of the Gospel. We have no verifiable evidence that the Mormon witnesses were actually appointed by Jesus.
3. Multitude Verification What the N.T. witnesses declared was easily confirmed by hundreds of hundreds of people (multitudes) who experienced the teaching of Jesus and saw His many healings (as well as experienced them). If the disciples were wrong in their witnessing the whole countryside would have confronted them.
The Mormon witnesses do not have this volume of confirmation by hundreds of people. They relate private meetings with J. Smith.
4. Continuous Witness What the N.T. Apostles (witnesses) declared was a message based on experiences over a period of 3 1/2 years. They had plenty of time to interact with Jesus...experience many spiritual events...see the miraculous deeds of Jesus on quite a few occasions.
Mormon testimony did not have nearly as much experience time. Nor could they confirm that the "alleged translation" of the alleged Golden Plates was an accurate one since they never spoke or wrote "Reformed Egyptian."
(Note: The idea that Joe Smith was able to carry off the Plates that were made of Gold is incredulous. No strong man could carry that much weight under his arm! Gold is extremely heavy!)
5. Cross Examination The N.T. Witnesses were cross examined in courts of law and in the palaces of kings (rulers). They were given the "third degree." Which means they were intensely grilled and tortured to see if it could break their testimony! None reneged.
Were the Mormon witnesses tried in a Court of Law? Were they cross-examined by attorneys and judges? As noted in other answers, Spaulding's relatives would have negated their testimony, if they ever were.
6. Supernatural Prophecy The Christian witnesses's message was confirmed by ancient supernatural prophecies, as Peter declared on several occasions. (Acts 2; 2 Peter) What they testified to was exactly what God had declared would happen.
What prophet, thousands of years ago, ever prophesied of Joe Smith and his Witnessers? What prophet ever mentioned "Golden Plates"? Like Peter said, we (traditional Christians) have a more sure word of prophecy!
7. Miraculous Healings The N.T. Apostles' message was confirmed by Jesus with "signs, wonders, healings, miracles" just as Jesus said would happen.
Which of the eleven Mormon witnesses ever "healed the blind eyes"? "Raised the dead"? "Caused the lame to walk"? Jesus NEVER confirmed the word of the Mormon witnesses, so it is reasonable to reject their alleged testimony.
8. Reformed Egyptian? We all know what printing and cursive writing is. The ancient Egyptians had the same type of writing: hieroglyphics and Demotive style. These do not indicate a "reformed" Egyptian language. In this day and age of Egyptology research, to appeal to and testify to such a language is quite suspect.
Also, the diagram and hieroglyphics that were inserted into The Book of Abraham of Mormonism, has been proven to be doctored by Smith, and shown to be fake! The ancient Sensen papyrus that Joe Smith was suppose to have translated as pictures of Abraham resurfaced in modernity. It was found in the Metropolitan Museum of Art...and accurately translated! Nothing about Abraham at all; just a piece from an Egyptian "Book of the Dead."
Conclusion For all of these reasons it is logical to reject the testimony of the Mormon witnesses. Their witness is quite suspect.
So how does the confirmed Testimony of the Twelve Apostles line up with the Mormon witnesses? It's out of the ball park! A home run for all to rejoice about! It's a winning team by any stretch of logic. The Mormon scenario doesn't even come close. We cannot "compare" the two classes of witnesses, but only "contrast" them.
This is why millions of Christian believers have confidence in their religion: it's based on irrefutable testimony that cannot be gain-said. What was testified to was not done in a corner!
See also Testimony of the Evangelists by Simon Greenleaf: https://en.wikipedia.org/wiki/Testimony_of_the_Evangelists and Norman Geisler's article "David Hume's Criteria for Witnesses", Encyclopedia of Christian Apologetics, and Evidence That Demands a Verdict by Josh McDowell, and the several books by Lee Strobel.