I am not that proficient on the subject but here is a clip that makes sense taken from here...http://www.matthewmcgee.org/2gospels.html
Peter's Gospel and Paul's Gospel
Both Peter and Paul taught that Jesus Christ was the Son of God, that he was crucified, and that he rose from the dead on the third day. So one might ask, "What is the difference between their two gospels?"
Earlier in this article, we discussed rather thoroughly the difference that Paul spoke to Gentiles and Jews whereas, Peter spoke to Israel only, with the one exception of the house of Cornelius.
A second key difference is that in making the offer of the kingdom to Israel, Peter spoke of the resurrection in order to show that the Lord was alive and could still return to be Israel's King (Acts 3:19-21). Christ's death and resurrection, the sign of Jonah, were stated as evidence. However, Peter was not proclaiming them as part of the gospel of the kingdom. But Paul taught the that the death and resurrection of Jesus Christ are essential parts of our gospel of grace.
A third difference is that Paul taught that Jesus Christ died as a sacrifice for our sins, and that we are cleansed by His blood. But in all of his sermons in the early chapters of Acts, Peter made no mention of this.
Decades later, near the end of their lives, Peter and John each wrote of the cleansing blood of Jesus Christ (1 Peter 1:2-3, 18-21, 2:24, 5:12 and 1 John 1:7 and 2:2). However, in the early parts of Acts, they never mention the blood, sacrifice, propitiation, or that Jesus Christ died for our sins. It had not yet been revealed.
Paul also told the Gentiles that Jesus Christ willingly gave up his life for our sins (Galatians 1:4). Whereas, Peter repeatedly accuses the Jews of murder. One example is Acts 2:36, where Peter says, "... Jesus, whom ye crucified ...." Peter also says in Acts 3:14-15, "But ye denied the Holy One ... and killed the Prince of life ...." Then in Acts 5:30 he says, "... Jesus, whom ye slew and hanged on a tree." Finally Stephen, who also preached Peter's gospel, told the Jews in Acts 7:52, "Which of the prophets have not your fathers persecuted? and they have slain them which shewed before of the coming of the Just One; of whom ye have been now the betrayers and murderers ...."
But Paul, on the other hand, constantly stressed the sacrificial nature of the death of Jesus Christ, "Whom God hath set forth to be a propitiation (appeasing sacrifice) through faith in his blood ..." (Romans 3:25). The blood of Christ is not mentioned by the Peter and the other 11 apostles in Acts, yet it is a vital part of the gospel of grace. One must conclude that either the twelve were negligent, or that it had not yet been revealed to them that Christ died a sacrificial death. Certainly the apostles, filled with the Holy Spirit, did not dispense an incomplete gospel, or those that heard it would have been without hope. So the sacrificial nature of Christ's death had not been revealed to them by God, just as we saw earlier in the section titled "The Mystery of the Gospel of Grace". In reference to the cross, Peter does explain in Acts 3:18 that "But those things, which God before had shewed by the mouth of all his prophets, that Christ should suffer, he hath so fulfilled." However, Peter does not link Christ's death to the justification of sinners.
It is of utmost importance to realize that Paul's letters are filled with the fact that the crucifixion of Jesus Christ was the sacrifice that paid for our sins. So we will allow a page or so here to look at a few example passages.
In Romans 5:6-11 Paul writes, "For when we were yet without strength, in due time Christ died for the ungodly. For scarcely for a righteous man will one die: yet peradventure for a good man some would even dare to die. But God commendeth his love toward us, in that, while we were yet sinners, Christ died for us. Much more then, being now justified by his blood, we shall be saved from wrath through him. For if, when we were enemies, we were reconciled to God by the death of his Son, much more, being reconciled, we shall be saved by his life. And not only so, but we also joy in God through our Lord Jesus Christ, by whom we have now received the atonement." So many people think that they have to clean up their life first before they can be saved. But this passage clearly shows that Christ did not die for the righteous, but "for the ungodly". Otherwise, His death would have been in vain, for Romans 3:10 says, "... there is none righteous, no not one."
In Ephesians 1:7 Paul writes of Jesus Christ "In whom we have redemption through his blood, the forgiveness of sins, according to the riches of his grace ...." Then in Ephesians 2:12-13 Paul explains how we were "... without God in the world: But now in Christ Jesus ye who sometimes were far off are made nigh by the blood of Christ."
Many other passages in Paul's letters emphasize this same point, including:
Colossians 1:20, "And, having made peace through the blood of his cross, by him to reconcile all things unto himself; by him, I say, whether they be things in earth, or things in heaven."
1 Thessalonians 5:9-10, "For God hath not appointed us to wrath, but to obtain salvation by our Lord Jesus Christ, Who died for us, that, whether we wake or sleep, we should live together with him."
Galatians 2:20, "I am crucified with Christ: nevertheless I live; yet not I, but Christ liveth in me: and the life which I now live in the flesh I live by the faith of the Son of God, who loved me, and gave himself for me."
Romans 8:31-32, "... If God be for us, who can be against us? He that spared not his own Son, but delivered him up for us all ...."
Romans 4:24-25, "... if we believe on him that raised up Jesus our Lord from the dead; Who was delivered for our offences, and was raised again for our justification."
1 Timothy 2:5-6, "For there is one God, and one mediator between God and men, the man Christ Jesus; Who gave himself a ransom for all ...."
Ephesians 5:1-2, "Be ye therefore followers of God, as dear children; And walk in love, as Christ also hath loved us, and hath given himself for us an offering and a sacrifice to God for a sweetsmelling savour."
2 Corinthians 5:21, "For he hath made him to be sin for us, who knew no sin; that we might be made the righteousness of God in him."
********************************************************************************************************************
You have to grapple with the prophetic program involving the Kingdom/Israel on the one hand and the Mystery/Body of Christ as revealed to Paul on the other. Two separate programs but both linked at Calvary.
My friend, Peter and Paul were teaching different people therefore they would approach them differently. The Jews were well versed in the Sciptures the Gentiles were not, therefore they would require different approaches in teaching