The topic area you have wandered into is commonly known as ‘The Parting of the Ways’ and is a subject of intensive and extensive scholarly interest. It’s the question of when, how, and why did Christianity become separate and distinct from Judaism. We’ve tossed this around a couple of times here on /r/AskHistorians. Here's two earlier threads on the topic here and here.

It’s really helpful to recognise that the question presupposes several things that you probably don’t realise. Firstly, the idea that religions are discrete ‘entities’ and that Judaism was a ‘thing’ and that Chrisitanity became another ‘thing’. What was 1st century Judaism? What were the borders of its religious ‘identity’? Josephus talks about 3 distinct philosophies among Jews: the Sadducees, the Pharisees, and the Essenes, but is also willing to talk about the political ideologies of the Zealots, as well as the political affiliations of Hellenising Jews. Are Jews who live like Gentiles still ‘Jews’? One of the great benefits for the Jews of their interplay of ethnic and religious identity was that Judaism appeared within the Roman Empire to be a religio with both ties to a particular geography and ethnos, as well as Antiquity, which was highly regarded.

If you accept that there is some level of basic veracity in the narrative of Acts, our only source that makes a decent claim to narrate something of the earliest years after the crucifixion of Jesus, his followers in Jerusalem continued to operate and practise within the sphere of ‘Jewish’ identity, gathering a reputation as a distinct group within those circles. However the movement’s outward focus, missionary and proselytising, quickly leads to the incorporation of Hellenistic-language and culture Jews, and then to God-fearers (interested Gentiles who had not made a commitment to become Jews), and eventually to Gentiles as Gentiles. Suddenly a host of theological issues arise, particularly over the relationship between Gentiles and Jews, the role of the Torah in this new Christian movement, and much of the New Testament corpus is taken up with these questions.

That issue, in many ways, creates an impetus that makes clear a distinction that was, if present beforehand, only implicit. Or to put it another way: the key questions of Jesus’ resurrection, the claim of Jesus’ messiahship, and the acceptance of the Gentiles, are 3 features where newly emergent Christianity differed sharply enough with the majority of other varieties of Judaism, that both groups began to distinguish themselves more clearly, deliberately, and publicly.

Did Jesus’ first followers intend to ‘found’ a new religion? I would answer no. Based on the documents left by the first and second generation of believers, that core of original followers conceived of their discipleship of Jesus in very ‘Jewish’ terms, and they would probably have liked to see the majority of Jews accept their view of Jesus, along with their emerging views on the Torah and related issues. However, the trajectory of beliefs that sees Peter take the initiative in accepting a Gentile into the new believing community, a trajectory extended by Paul to an unprecedented degree, is seen as the logical, faithful extension of what they learnt from Jesus. The NT documents do not portray this emergent ‘Christianity’ as somehow a betrayal of Jesus’ “Jewish religion”, but the proper expression and fulfilment of it.

On the other side of the equation, the destruction of the Temple and razing of Jerusalem in 70AD radically altered the path of Judaism. It’s from that point onward that you get the emergence of Rabbinic Judaism. A Judaism without the Temple must reformulate and function differently. Furthermore, the political instability, compounded by the subsequent Jewish war and overwhelming opposition from the Roman state, made former ‘versions’ of Judaism untenable. Rabbinic Judaism emerged as the surviving form of Judaism, and itself defined itself in new ways, none of which appear to be very accommodating of the core beliefs of Jesus’ followers in the centuries that follow.

This is precisely the sort of thing that the folks at /r/AcademicBiblical know and are trained to answer.