The distinction emerged in the later half of the first century and was more or less complete by the end of the Bar Kokhba revolt in 136. Internally, Paul's emphasis on Gentile converts not being required to adopt Jewish customs held significant weight and eventually became the dominant view within the movement. /u/TruePrep1818 mentions some of this in his response.

Externally, the Roman Jewish Wars were very important. By the time of the First Roman-Jewish War in 66-73, the difference was likely just beginning to emerge. But the early Christians largely took the destruction of the Temple in Jerusalem as evidence that the end times predicted by Jesus were beginning (this view is articulated in the Gospel of John, written around 85-95), which they took as proof of his teachings. And like nearly all people in this time period, the early Christians interpreted the Jewish military defeat to the Romans as evidence that God had abandoned them - in their minds for failing to recognize Jesus as the Messiah. Post-war bans on circumcision and the imposition of the "temple tax" on the Jews to Rome also contributed to the widening difference. Christians likely began to emphasize their non-Jewish character in an attempt to exempt themselves from the tax, but I'm not sure we have a written source for this claim.

During the Bar Kokhba revolt in 132-136, the Jewish rebels slaughtered Christians alongside gentiles when the Christians declined to participate in the revolt. After the rebellion was crushed by the Romans, Emperor Hadrian attempted to annihilate the Jewish religion entirely, as he was convinced Jews could never coexist peacefully within the empire. He banned the teaching of Jewish religious practices, banned Jews from entering Jerusalem, and scattered (or killed) the majority of the Jewish population in Judea. He even removed the province of Judea from the map, combining it with Syria to form Syria Palaestina. By this point, its fairly safe to say the Christians viewed themselves as entirely distinct from the Jews, after having suffered at their hands during the revolt and eager to emphasize their new Christian identity to convince the Romans they were nothing like the Jews.

edit: spelling

I can't claim to be a genuine expert on this, but it was something discussed in the Yale Open Course I watched on the New Testament. Basically, the first three Gospels (Mark, Matthew, and Luke), are more emphatic about the Jewish identity of Jesus, with Matthew in particular being focused on it. With the Gospel of John, the final canonical Gospel written around 90 AD, we begin to see Jesus' Jewish identity being minimized and the Jewish community even beginning to be maligned within the Christian consciousness. In other words, while there isn't a hard, fast date for when the Christian Church officially "broke" with Judaism, we can see Christian identity forming outside of the Jewish stream during the later half of the First Century when John was writing.

The divide between Jews and Christians also has a great deal to do with the ministry of Saint Paul, who believed it was his mission to spread the word of Jesus to the gentiles. The Apostles of Jesus, for the most part, confined their preaching to the Jewish communities that they were a part of. Paul's brining gentiles into the Church allowed for a Christian identity to form outside of Judaism.

I'm sure someone more knowledgable than me can supplement and improve on this, but I figured I'd at least try my hand at giving an answer.

As a tangentially related question, when is the last record of a self identifying Jewish community of Judeo-Christian theology (philosophy?) with continuous roots. That is, not a Jewish Community that adopted Christianity or Christian practices after there was a distinct Christian community, but one that adopted early Judeo-Christianity and was one of the last we have record of to be absorbed into greater Christianity or its record lost.

Does that question even make sense? If so, how should it be phrased so it's not as cumbersome?