What is the difference between God and the Spirit of God?
There is a related question on this asking, "Is the Holy Ghost the same with the Holy Spirit".
Question: "What is the difference between the Holy Spirit and Holy Ghost?"
Answer: Of the modern English translations of the Bible, it is only the King James Version of the Bible which uses the term “Holy Ghost.” It occurs 90 times in the KJV. The term “Holy Spirit” occurs 7 times in the KJV. There is no clear reason as to why the KJV translators used Ghost in most places and then Spirit in a few. The exact same Greek and Hebrew words are translated "ghost" and "spirit" in the KJV in different occurrences of the words. By "ghost," the KJV translators did not intend to communicate the idea of "the spirit of a deceased person." In 1611, when the KJV was originally translated, the word "ghost" primarily referred to "an immaterial being."https://www.gotquestions.org/Holy-Spirit-Ghost.html
There's a lot of similar sources on different Christian site that had answered this question and unanimously answered that the "Holy Ghost and Holy Spirit", are the same, they are the Third Person of the Holy Trinity.
Another source to read is here : Is there any difference in meaning between "Holy Spirit" and "Holy Ghost"? text**
We can see, that your question if we look at all of the answer cited or referenced, the Spirit of God, is the Holy Spirit and the Holy Ghost. But ofcourse, there are passages in scriptures that is not clear like the Book of Wisdom, Sirach and Proverbs and some passages in New Testament like the one you cited in 1Corinthians2-10-13.
So, how can be view this in light of the scriptures? One way to look at this is by using "science". Yes!, science can aid us in discovering our faith when passages in scriptures seems difficult to decipher. How can we visualize, the Holy Trinity, a "Spirit united as One" with the "Spirit of God" in a diagram?
For the sake of clarity, we can see the God, a Spirit, "breath out a "spirit" to Adam to bring divine life to him. In contrast, Jesus upon resurrection also "breathed out" a "spirit" to all the Apostles to restore back "divine life" to them.
The question is, is the spirit that God the Father and Jesus breathed out is the "Holy Ghost or the Holy Spirit", the Third Person in the Holy Trinity" or it is just a "spirit that emanates from them"?
If we accept, that the "spirit" that God the Father and Jesus breathed out, is just a "spirit" that emanates in their Spirit as God, then it will be helpful in understanding the three difficult passages in Proverbs8:22, Wisdom chapter 7 particularly Wisdom7:25 and Sirach chapter1. All this passages pertain to "Wisdom" a "spirit that was created or emanates from God, and it was created.
Let's look at Wisdom7:25. As it describes that a Spirit or Wisdom emanates from God, and this is the same Spirit that God "breath in" to Adam to give him divine life.
- For she is a breath of the might of God
and a pure emanation of the glory of the Almighty;
therefore nothing defiled can enter into her.(Wisdom7:25)
Book of genesis described, God breath into Adam nostrils, God breath in him a "spirit" and this emanates from God.
Then the LORD God formed the man of dust from the ground, and breathed into his nostrils the breath of life; and the man became a living person.
And the gospel proving that Jesus is God or the one with the Father, they are co-substantial, after His resurrection, had shown that like the Father, Jesus can also "breath out" the "spirit" thru His own power.
When He had said this, He breathed on them and said, “Receive the Holy Spirit. (John20:22)
Therefore, we can see that both the Father and Jesus breathed out the Spirit that brings "divine life" to Adam and restore it back to the Apostles.
The confusion arises, in interpreting Proverbs8:22 and it's parallel passages in Sirach1:7-9
“The LORD created me at the beginning of His way, Before His works of old.
Before all other things wisdom was created;
and prudent understanding, from eternity.†
The root of wisdom—to whom has it been revealed?
Her subtleties—who knows them?
There is but one, wise and truly awesome,
seated upon his throne—the Lord.
It is he who created her,
saw her and measured her,
Poured her forth upon all his works,
upon every living thing according to his bounty,
lavished her upon those who love him. (Sirach1:4-10)
And in the Book of Wisdom chapter 7 it was described this way, it pertains to a "Spirit" that emanates from God.
Nature and Incomparable Dignity of Wisdom
For in her is a spirit
intelligent, holy, unique,
Manifold, subtle, agile,
clear, unstained, certain,
Never harmful, loving the good, keen,
unhampered, beneficent, kindly,
Firm, secure, tranquil,
And pervading all spirits,
though they be intelligent, pure and very subtle.
For Wisdom is mobile beyond all motion,
and she penetrates and pervades all things by reason of her purity.
For she is a breath of the might of God
and a pure emanation of the glory of the Almighty;
therefore nothing defiled can enter into her.
For she is the reflection of eternal light,
the spotless mirror of the power of God,
the image of his goodness.
Although she is one, she can do all things,
and she renews everything while herself perduring;
Passing into holy souls from age to age,
she produces friends of God and prophets.
For God loves nothing so much as the one who dwells with Wisdom.
For she is fairer than the sun
and surpasses every constellation of the stars.
Compared to light, she is found more radiant;
though night supplants light,
wickedness does not prevail over Wisdom.
Now all this biblical passages can be simply put in a diagram, using science to represent the Holy Trinity like the three isotopes of Hydrogen atom, and the electron orbiting represent the "spirit of wisdom" that emanates from God.
Also, this illustration does not comes from Catholic Church, it is my own scientific way of representing the Holy Trinity with the spirit of Wisdom emanating from them. And take note, the Father represents the proton with a positive charge or Will unlike the Logos and Holy Ghost which represented the two neutrons because they will be both sent with "no authority" on their own. And the electron represents the Wisdom, that simply possess the power like the Omnipotent God, the electron or spirit is not God, but since it is all-knowing, the spirit is just an emanation or a reflection of God.
Refer again to the passage describing the "Nature and Incomparable Dignity of Wisdom" in book of wisdom chapter 7 as described by King Solomon.