In Acts, the conflict between Paul and Barnabas seems to pop up out of nowhere. They have just had a successful missionary journey. They have just prevailed at the meeting in Jerusalem. Now suddenly they have an argument over whether John Mark should go with them on a second missionary journey?
The book of Galatians suggests that the argument was over more than just John Mark. In the disagreement at Antioch (Gal. 2:11-14), Barnabas ends up taking Peter's side over Paul in the dispute. While it would be convenient to think that Acts 15 resolved this dispute, it seems more likely that Paul lost this argument and that the tensions lingered on for a number of years.  Paul makes it a point to tell us earlier that he convinced Peter, James, and John about circumcision (Gal. 2:6-10). But he can't tell us that he convinced Peter and Barnabas at Antioch, because he likely didn't.
What was Barnabas thinking? Didn't he agree with Paul that Gentile believers did not need to be circumcised? Had he not eaten with Gentile converts until the blow up at Antioch?
Perhaps he agreed with James' logic. Yes, Gentiles can escape God's wrath without following the Jewish purity laws, being circumcised, and so forth. But that doesn't mean that Jewish believers can stop keeping the purity laws. Unless the Gentiles are willing to follow certain rules, the two will not be able to eat together.
We can't read Barnabas' mind. Maybe he sided with Peter for reasons of authority. Maybe he thought it was important to submit to the authority of Peter and James, even if he disagreed with them. Maybe he was being pragmatist. Maybe he thought it was a good tactic to yield for the moment until a compromise position could be worked out.
For Paul, this was perhaps the moment God used to sow the seeds of justification by faith. He knew the Gentiles were "in" without having to be circumcised. How quickly did his theology explaining this mysterious fact follow? His thinking apparently was still forming back when he was in Galatia, for someone was able to convince them that he preferred that they get circumcised (Gal. 5:11)!
But this event in Antioch would seal the deal. No, these sort of "works of Law"--things like purity laws and circumcision--must have nothing to do with a right standing before God. Had he not been blameless at keeping them (Phil. 3:6)? If he had not been right with God through them, then no one could get right with God through them. Justification must be completely through the faithful death of Jesus, something we put our faith in when we are baptized. Then God gives us the Spirit as his seal of true ownership.
Acts remembers John Mark as the flash point, but he was only the symptom of a deeper conflict. Had John Mark been the one who tattletaled on Paul's mission to Gentiles when he went home early from the first missionary journey? After all, he was from Jerusalem and well connected to Peter and the inner circle. If all we had were Acts, we might think it was just because Paul thought him a quitter, and maybe that was part of it.
But God redeemed the conflict. God used it for good. The result was two missions instead of one. If they had agreed, Paul might merely have gone back to Cyprus. But now he and Silas set course to spread the good news to new lands. They will not only presumably take Titus with them, but they will add a young man named Timothy to the ranks of missionaries (Acts 16:1-3).
The rift was not permanent. Paul speaks in positive terms of Barnabas in 1 Corinthians 9:6, and he speaks positively of Mark in Colossians 4:10. Christians will not always agree on everything, and there is a time to be unified in disagreement, to agree to disagree. That is exactly what Paul and Barnabas did.
 If, as I think likely, Galatians 2:1-10 is Paul's version of Acts 15, then the incident at Antioch actually happened after the meeting in Jerusalem. I of course think Galatians fits better in the middle of Paul's ministry rather than at the beginning.