The gospel of Matthew and Paul's letters reflect close but different understanding of how to relate to Judaism. How should Christians relate to Jews?
This is a very complex question, but it is excellent. Here are two points that will help you get started.
First, the gospel of Matthew is the most Jewish of the gospels in the New Testament. The author is at pains to show that the fulfillment of the Hebrew Bible is the emergence of Jesus. In other words, he is the long awaited Messiah, the anointed one of God. This is why Matthew constantly says that the Scriptures were fulfilled in Christ. Hence, we can conclude that Matthew is optimistic that many Jews would embrace Christ. One religion leads to another. There is a sense of hope.
Second, in Paul's letter, it is clear that there is a break between the two religions. Paul is less optimistic. Part of the reason for this is due to the fact that many Jews have rejected his message of Christ. The many persecutions point in this direction. To be sure, Paul wants to convert Jews, but he is realistic that Christianity is a new religion and that there will be opposition. So, he calls Christians to live in peace, if they are able, but he also calls them to stand firm in the faith in view of the resurrection of Christ - 1 Corinthians 15:55-58.