Just to amplify Alkalannars point a tiny bit. Consider even just the title 'sermon on the mount', with where lessons from the Torah would typically have been recited (is 'sermon' anachronistic?) and jewish religious rituals would have been held (mostly) in the temple and later centered around synagogues. This vividly illustrates some of the origin of the differences between them, like their moral codes.
To complicate things slightly some believe and have argued persuasively that Jesus was a Jew, in books like 'the Passover Plot' and would have considered himself such, and not the first Christian, passing that mantle onto Paul or his brother James. Many books fairly critical of traditional christian views were quite common and in wide circulation by the 70s (maybe in part because of the 6 day war and OPEC embargo?), like the Last Tempation of Christ (more a pure literary work of fiction, but it also raises some interesting questions/issues).
There are suggestions that the early group centered around him and his teachings split into two sects close by or soon after his death, between those who considered themselves Jews (like his brother James in Jerusalem), and those who felt they didn't have to follow all the old Jewish laws and practices, in effect a new religion. A (the?) major champion of the latter was Paul, who spread his new gospel to the Gentiles, and thus germs of Christianity (as you may know it) was born, and flourished away from the holy land, the circumstances and people who gave it birth and (arguably) those who were closest to Jesus. To complicate things even further that separation was exacerbated (if not directly caused) by the first Jewish Roman war, the great revolt. This offers a further possibility, that surviving rebels and/or their sympathizers being scattered to the winds (some as slaves?), in their flight from the holy land, dispersed into the wider roman world, concocting a new religion to explain away their failure and giving them a new mission. There has since been questions about who precisely his followers were (e.g. was Mary Magdalene an apostle?), was he married and who his family was (besides just Mary and Joseph), did he have children (e.g. Holy Blood, Holy Grail and the DaVinci Code), what about lost Gospels (Nag Hammadi) and probably others, besides more traditional theological concerns.
It's not hard to speculate this sort of historical context could easily have caused many of the earliest christian heresies like Arianism in Alexandria, most centered on the precise nature of Jesus and the trinity, Christological questions like was Jesus a man or god? This sort of thing was very important because they felt they had to somehow explain things to people of a rational bent (like Greeks and Romans), that could otherwise appear to be contradictory, like God crucifying himself. Maybe not entirely coincidentally, this is about the time many of the oldest schools of philosophy started closing.
Perhaps sects in or nearer the holy land (e.g. Antioch, Alexandria with things like Nestorianism and Adoptionism) dogmatically clung to older traditional beliefs perhaps surviving largely unchanged from earliest times when they were among the first converts, maybe far closer to pure Judaism or where there may have been surviving memory, history or traditions concerning any real people mentioned in the Gospels (where maybe people who might have seen Jesus on the mount or known him emigrated to). These were perhaps controversial ideas like that Jesus was actually just an ordinary man (carefully walking the line between being an influential a teacher and a dangerous political leader), while those further flung Christians came to develop and believe more radical and revolutionary theologies, which came to dominate (maybe after converting major groups of new converts like in Rome). The demand for standardization, as Christianitys influence waxed in the Empire would have been a strong force for standardization and could have overwhelmed smaller holdouts.
The important point for you however, is concerning how Christians of much later traditions view the Sermon on the Mount, and how they compare to traditional Jews views. Of extreme importance was the nature of 'messiah' in Judaism, and Christianity, and is arguably one of their greatest divergences with major consequences for their later development. There was a few Jewish notions of the Messiah (which may have changed) where they believed either he hadn't come yet, or that he came but was Judas Maccabeus, among others. A Jewish Messiah was a far more worldly thing, since their Babylonian exile (a major defining moment for the religion like their Egyptian exodus) basically a new King of Isreal, who would restore the Temple and line of David on the throne of the Jewish nation (IIRC). Later this became more political, anyone who would free the Jews fiear from the Greek Rule (arguably Revelations), then the Roman yoke, and then maybe from Islam, taking you pretty much to the present. The Christians by contrast abstracted the Messianic mission very early on, as a sort of spiritual rescue and restoration, not a worldly political or physical one. This might have been terribly convenient, after you've seen your leader gruesomely capitally punished, somehow making an abject failure into a success. Anyway, the basis for answering what Jesus did, what his moral lessons (the parables) meant, to a great extent depends on who you believe he was, which Christians and Jews disagree on.