To my knowledge, there is only one disagreement between Peter and Paul. Paul mentions it in Galatians 2:10-14. When Peter visited Antioch, and the Christians there, even though he was the first apostle to witness to the Gentiles at Cornelius’ house (Acts 10), he avoided the Gentile Christians because of the Judaizers, Jewish teachers from Jerusalem.
Paul saw this as an affront to the gospel. The gospel teaches that Gentiles and Jews who know and follow Christ no longer care about their ethnicity. Any squabbles or disagreements with one another are subservient to Christ and his message and kingdom.
As he says in Galatians, there is neither Jew nor Gentile (Galatians 3:29?). So Paul saw Peter’s avoidance of the Gentile Christians at Antioch, of which there were many, as a challenge to the gospel message.
Peter essentially either out of fear of these Jewish visitors from Jerusalem or some other reason, as siding with them on the matters of Christians needing to be circumcised (a big matter brought up throughout Galatians) and a number of other issues the Jewish teachers would raise.
From what we can gather in Galatians, these Judaizers followed Paul around and tried to catch him in teachings he did not teach. They tried to catch him on circumcision, following the law of Moses, and Christian libertarianism.
Galatians was written before the church decided the matter on circumcision and how much of Judaism a Christian Gentile would have to follow in Acts 15. Galatians may be the earliest written of the New Testament books.
So these matters had not yet been settled at this time. This is why Peter did not have clear direction on whether or not to cater to these Jewish teachers. After all, he was called the missionary to the Jews while Paul was the missionary to the Gentiles.
We see other squabbles between the Hellenized Jewish widows and the widows from Jerusalem in Acts 6. There was still a lot of tension between the Gentile Christians and Jewish Christians in the time Galatians was written.
It’s no surprise that because there is no clear direction from the church on this matter yet, Peter and Paul would come to odds about the issues. If Paul would have maintained the teaching of these Jewish teachers visiting Antioch, his ministry would be short-lived.
Gentiles tended to be, “God-fearers” who honored the customs and practices of the Jews without going through circumcision handful proselytization. If Christianity became Judaism 2.0, they would have the same problem with Gentile believers.
Thankfully, the church decided in Acts 15 that Jesus was enough for the Christian religion. There was only a need to follow four rules given to Noah after the flood in the beginning of earth once again. These four rules happened even before the laws of Moses were given by God on Mount Sinai.
So the disagreement between Peter and Paul was significant at the time, probably one of the key factors driving Paul to go to the Council and present his case before the church for consideration and ruling.
This allowed Christianity to expand not based on ethnicity but on the gospel message. I don’t want to say Paul won the argument. I want to say the Holy Spirit helped everyone involved to understand the further implications of the issues at hand. The church listened to the Holy Spirit and we have the gospel message alone for consideration in conversion.