Three really good resources I'd like to point you to to understand this delineation are the Bible itself, Christianity and Rabbinic Judaism by Hershel Shanks and Border Lines by Daniel Boyarin.

To condense the very complicated theological and historical mess that comprises volumes, the clear distinction between Christianity and Judaism emerged somewhat gradually and over time. There were numerous Messianic claims in and around the time of Christ's life (or, for the skeptics, the time period where Christ was said to have lived); additionally, there were some major political, social, and religious changes that happened to unfold for Judaism around that time, thanks in no small part to the nascent Christian movement.

First, let's address the comment below, which I will preserve in case it is removed:

I am more interested in why people say Jesus was a jew in the first place. I concede that he was "raised jewish", but if he "made himself the Son of God" (John 19:7), and believed in his own divinity, then he was a Christian.

Isn't his belief in his own divinity the litmus test here?

The answer to this question depends on when it's being asked. Today that is absolutely part of the litmus test, as Jews today reject that the Messiah will be God, the son of God, or divine in that way. Back then, it wasn't really a settled question. Cyrus the Great could have been the Messiah because of his instrumental work in rebuilding the Temple--There are claims that Herod, John the Baptist, and others are the Messiah for varying reasons. Over time, as Christianity moves a long, Boyarin argues that this conception of the Messiah becomes absolutely critical in the separation of Judaism from Christianity very early on.

There's also a political component. In Acts, we see that Christianity becomes opened up to gentiles. At around this time, subjects of the Roman empire are required to bow before the Emperor or his likeness, but the Jewish people enjoy an exception. Christians are also desirous of this exception on fairly identical theological grounds--in fact they still claim to be Jewish! The exception is not granted and this is a significant source of early Christian persecution. Christians being otherized from Jews by Roman authorities is part of how this divide begins to grow. Jews and Christians start examining what makes them different more and more.

Then we have the destruction of the Second Temple. This forces Judaism to transform radically. Synagogues already existed at this time, but now with the Temple gone, they become the primary centers of Jewish life. The Epistles of Paul paint a picture that Synagogues are already a huge part of Christian life before the destruction of the Temple. It's this event that really sets into motion the creation of Rabbinic Judaism that we know today, and the Pharisees, Sadducees, Essenes, and the so-called Fourth Philosophy are subsumed into this larger Jewish identity.

Christianity is also experiencing growing pains, and will continue to for the next couple hundred years. Between the destruction of the Second Temple and the First Council of Nicaea, Rabbinic Judaism and Christianity are really growing alongside each other and defining themselves against the backdrop of one another.

This legacy is most apparent when you compare the extant Byzantine Liturgy with the modern Orthodox and Conservative Jewish liturgies. The parallels are hard to miss. By the time Constantine decriminalizes Christianity it is a completely separate religion from Judaism. I think the story of how the Samaritans and Jews drifted apart during the Babylonian exile is very instructive too. Both faiths have the same origin, and both see themselves as being the successor to that legacy.