The Big Split - Inclusion of Gentiles
If a Gentile wanted to follow the God of Israel, the accepted method among the Jewish rabbis was that the person convert to Judaism. This involved becoming circumcised (if male), being immersed, and taking on the obligation to keep Torah, thus becoming Jewish and coming within the covenants God had made with his people.
When the apostles decided in Acts 15 that Gentiles could become part of God's people as Gentiles, without undergoing ritual conversion (circumcision), it was a major slap-in-the-face to Jewish teaching. Even though the apostles conclusion was supported by the Tanakh, which Paul argues in Romans 15, it was in opposition to the teachings of the rabbis. This decision was the primary wedge between the apostles and the Jews that grew into the split in the years following.
Paul argues that the inclusion of the Gentiles is at the heart of the gospel. He refers to it as the "mystery of the gospel" (Eph. 6:19), and elaborates further that it was something not previously revealed (Rom. 16:25; Eph. 3:3-9; Col. 1:27). But now, through the teaching of the apostles, it's being make known.
Paul symbolizes this teaching in Romans 11 by an olive tree, a familiar reference to Israel (Jer. 11:16; Hos. 14:6). Gentiles are pictured as wild olive branches who are grafted into the tree, though remaining wild olive branches. The people of God, though Jewish at its core, would be expanded to include those from all nations who turned to God and trusted in Jesus. (This refutes the belief that Gentile believers are really the lost tribes of Israel.)
It is interesting that when Paul met opposition from Jews in his ministry, it was invariably because he was including Gentiles. For example, in Pisidian Antioch (Acts 13:14-52) he received a warm reception to his first speech in the synagogue. But on the next Sabbath when hundreds of Gentiles crowded into the synagogue to hear him, the Jews became jealous and rejected Paul's message.
Gentiles were also the cause of Paul's final arrest in Acts 21:27-36. Some Jews from Asia saw Paul in the temple, performing a temple ritual. Since they had previously seen him in the city with Gentiles (and knew that he was teaching Gentile inclusion), they assumed that he had brought them into the temple. They assaulted him and beat him until he was rescued by the Roman troops. From there until the end of Acts, this group of Jews accuses Paul of opposing the temple and the Torah, charges which Paul vigorously refutes many times.
It is interesting that the Jews did not seem to object when Paul was preaching Jesus as Messiah. They only objected when he began to include Gentiles among those he was calling to repentance. This act threatened their favored status and sense of chosenness. It hit at the heart of their sense of identity. Because of this they began to oppose and persecute the followers of Jesus who taught this.
The Big Split - Fall of Jerusalem