At what point did Judaism and Christianity diverge? Christianity and Judaism share the Old Testament (The Hebrew Bible), so it should be safe to assume they are the same people at least during the Old Testament period. Since when did they disagree?
tl;dr> It began in the Council of Jersualem (55 AD), but was cemented by the destruction of the Temple in 70AD.
1. Gentiles were released from Jewish custom
The divergence is clearly articulated in Acts 15 - at the "Council of Jerusalem," often pegged at 50 AD - roughly 20 years after the Crucifixion. Acts 15 sets up the situation as follows:
In a nutshell - it's the old circumcision debate: Do converts need to go under the knife?
Prior to this disagreement, Christians were simply Jews that believed Jesus was the promised Messiah. When the Messiah came, however, he expanded His mission in such a way as to afford salvation to the Gentiles. (See Matthew 28:19-20)
Peter speaks to this issue, recalling the fact that the Gentiles were being added to the mission - first in Acts 10, and now again in Acts 15:
That they were included, however, necessarily raised the question of whether or not converts needed to follow all the practices of the Jews. Peter, Paul, Barnabas, and the other apostles were clear that the grace of God implied that Gentiles need not do Jewish custom.
It was even suggested that a letter should be written, in which they said:
By exempting the Gentiles from the rest of Jewish practice, a separate identity began to form.
2. The Persecution of the Jews diminished their importance in the Church, and vice versa
With the destruction of the Temple 15 years later, it simply became much harder to become a Jew. As OzTorah writes:
With this polarization, rapid growth outside of Israel (Paul's ministry, in particular, was centered in Turkey and Greece, as evidenced by the fact that Corinth, Galatia, Ephesus, Phillippi, Colossae, and Thessaly all got their own books of the Bible!), and the persecution of the Jews inside of Israel, led to an imbalance of population.
Basically, the "Jews" dwindled and the "Gentiles" grew. That, coupled with the freedom from the practices of the Jews eventually led to a situation where the church's early identity divested itself of its Jewishness.
3. By the second century, there was hostility
With the firm acceptance of Paul's letters by the end of the first, the anti-Judaizer faction became well entrenched. By the middle of the second Century, some heretics like Marcion were already trying to remove any vestigial Jewish influence. When Marcion produced his first canonical list, for example, the entire Old Testament was relegated to non-canonical status. (Note - Marcionism is considered heretical nowadays, and rejecting the Old Testament is considered heretical, but I point it as way of evidence that the split had already occurred.)
By this time, "Christ-killer" rhetoric and the like, along with regional chauvinism prevailed, and enmity between the Jews and the Christians was mainstream. Modern Judaism took root in the Pharisitcal and Essene movements and followed its own course thereafter.
That's fairly straightforward.
Jesus Himself was Jewish. The divergence came from Him. Whether it was at His death, burial, and resurrection, immediately afterward, or when He began His ministry is up to interpretation and debate, but it was Jesus, and the New testament (New Covenant) established by Him that marked the split.
Those who recognized Jesus as the promised Messiah were the first Christians. Perhaps they could have been considered Messianic Jews, but since Christianity is defined as following Christ, following Him is the logical determining factor as to whether or not one is a Christian. Jews who followed him were, for all intents and purposes, the first Christians, and that's where the divergence lies.
As David has pointed out, but without detail, the division was with Jesus' life, death, and resurrection. There was nothing that could be called Christianity before Jesus.
During the last three years of His life Jesus gained many followers, however, he never made any deliberate attempt to divide the Jewish faith, nor change it.
These are Jesus' own words. He was claiming that the Law (the ancient Hebrew texts) was not to be removed by his coming, but to be uplifted and proven and fulfilled. Jesus was claiming to be a fulfillment of prophecy. The Gospels (Matthew, Mark, Luke, and John) detail Jesus' ministry during the last three years of his life, where he attempted to fulfill Jewish prophecy as the Messiah or the Christ.
However, the Jews of the time rejected Jesus and had Him executed, but that was part of the prophecies. He was disgraced and his followers dispersed. This is also detailed in the Gospels.
Later, however, Jesus' disciples and a few other persons began claiming to have seen Jesus alive. Not alive in that He actually survived execution, but alive in that God had raised Him from the dead and glorified Him. Over the course of a short time many more claim to have seen the resurrected Jesus until He ascended into heaven 40 days (I think) after His resurrection. This too is detailed in the Gospels.
Most theologians would say that it is was this point that Christianity split from Judaism. There were still many Jews who would not believe the resurrection, therefore, there was nothing miraculous about Jesus, but many others did believe, therefore, He became known as the Messiah and the Christ, the One who fulfilled prophecy. Secular historians must neglect the resurrection as fact, however, they do not neglect that Christianity was recognized as from Judaism but different and distinct from Judaism quite early; most estimate that Christianity was well established as a separate religion before AD 200.
According to the Bible the word Christian, which means literally 'small Christ' and means figuratively 'strives to be like Christ', was used very soon after the Resurrection.
The Book of Acts was written to cover the events immediately after the Resurrection to show how Christianity came to be such a large movement then, which continues today.
Originally, Christianity, as taught by Christ (Jesus), was meant to support and augment Judaism. The Christ came to fulfill the Law. Christianity became something separate from Judaism very shortly after Jesus' Resurrection. The disagreement was whether He was the Christ or not. Christians believe He is and the Jews believe He was not. Because there were many on both sides the two factions were made. The factions persist today for the same reasons, generally speaking.
In the view of those who follow Christ, Christianity is a continuation and fulfillment of the Judaism of the Old Testament. A certain part of Judaism diverged from that lineage most prominently at about noon on the day Jesus was crucified:
Many other Jews, however, did recognize and follow the awaited Messiah. For example, the Letter to the Hebrews was directed to Jewish Christians. This is what the word "Christ" means: Messiah. So "Messianic Judaism" is effectively just "Christianity".
Alternatively, the point at which this divergence occurred was the moment when Jesus was conceived by the Holy Spirit, or when the first or last of the various Messianic Prophecies was fulfilled.