Christians often hold the misconception that “Jews know the Old Testament really well.” Not so. First, many modern Jews know more about cultural Judaism, some of which includes reference to biblical events, but much of which is largely 20th C information. As Frank Schwartz used to say “I just know about Delancey Street. I rely on you to explain why we do the things we do.” He exaggerated and was being too modest, but you get the point.
Second, even among those who are knowledgeable about the texts underpinning their faith, most Jews learn Torah and Talmud, not the histories and prophets. Not that they have anything against the histories and prophets, but Talmud takes a more elevated spot, and you can’t emphasise everything. There have been Jewish sects which would neglect the later OT books altogether, holding only to Torah, and in response, more dominant strains of Judaism made efforts to insist that some be retained in synagogue worship. But there is nowhere near the emphasis on the remainder of the Old Testament that one finds in Christian churches.
The use of wisdom literature is more mixed. The Psalms are still used in worship and personal devotions. Proverbs has been much-replaced by Talmud. Individual sections of wisdom literature are used for various holidays. But as with the rest of the Haftarah, there is an underemphasis to Christian eyes. Yeshivas teach Torah-Talmud.
Apart from any specific verses or sections of the histories and prophets which Christians have thought important to retain with emphasis, the development of Talmud – fascinating story in itself, BTW, though I won’t go into it here – contrasted with retaining the centrality of the prophets, creates a very different picture of God’s revelation. To overstate for emphasis, 1st Century Christians dropped Torah commentary, the Jews dropped the Prophets. The Christians quickly emphasised the entire illustrative and prophetic arc of Israel’s history, on into the New Covenant. (This was not a later addition. Jesus himself speaks this way, and his followers relate this narrative arc of creation, selection, and repeated rejection of prophecy from the very first.) The Jews moved to a different abstraction: now that the Temple has been destroyed, what is the remaining core and how do we express it?
I think this is an enormous chasm. To Christians, this looks like frank evasion by the Jews to not bother much about those places in their history where God told them they were sinning badly. That they cannot endure Isaiah and Habakkuk virtually guarantees that they will not hear Jesus. To Jews, this is a convenient reinterpretation of scripture by Christians because it could be made to point to their purported Messiah. In their eyes, Jews did the hard work of distilling their faith to abstract essentials while retaining concrete expressions - while the Christians went off on one of those wild-goose chases about a Messiah, which happens in Judaism from time to time. When we looked it all over, we decided that Torah had to be absolutely central, and understanding it in the new context our most important task.
This leads to cross-purposes in discussion, with Christians quoting Micah and Jews either focusing on Torah and its place in Judaism, or more often, wanting to discuss the history of the Jews in the Common Era. Neither rejects the other's focus as irrelevant, but considers it of secondary importance.
We identify break points in retrospect, because they provide neater, easier explanations. The destruction of the Second Temple was certainly a dividing line. Yet I think there are evidences of the break before that. The schools of Oral Law, or Mishnah, were certainly underway before 70 CE – the destruction in fact prompted Jews to write much of this down and organise it. Perhaps Mishnah can in turn be seen as a response to the defiling of the Temple in the days of the Maccabees – a prefiguring of its later destruction – a beginning of the process of distilling Judaism to Torah and commentary. All three major Jewish groups of Jesus’s time, Essenes, Sadducees, and Pharisees, show this movement away from later OT writings, the prophets in particular. (The Essenes, whoever they were*, seem to have kept some focus on the histories, and more messianic themes as well.) This is in great contrast to John the Baptist and to Jesus, who frequently refer to the prophets and histories.
If this split were already underway, it provides an interesting undercurrent to a few sections of scripture. When the Pharisees challenge Jesus about Sabbathkeeping, he answers from the histories, of David’s freedom to eat the consecrated bread in circumstances that would look unpermitted according to current rules. He quotes Hosea, refers to Jonah and Solomon; He refers to "this generation," as prophets repeatedly did, but is uncommon in Torah and Talmud, which stress the permanence of laws and rulings for Jews of all times; Matthew identifies this as a fulfillment of Isaiah.
When John asks Jesus if he really is the awaited one, Jesus answers from the prophets. Even when referring to the Pentateuch, Jesus refers to Moses and to Abraham – to the covenant figures – and expresses quite clearly that he a) will supersede the law or sometimes, b) already supersedes it. Commentary on the law he regards as less than unimportant, but actually pernicious. Even when referencing the Ten Commandments, he uses the odd phrasing “You have heard it said…” - not the usual emphatic endorsement that this is eternal command, but something milder. But as he takes the principle of the commandment against murder and extends it tenfold in the next breath, one can hardly accuse him of speaking against it. More probably, his point is that even the Commandments were training wheels, soon to be discarded. That the training wheels should themselves require smaller training wheels was to go in precisely the wrong direction.
*Current best guess: more a generic movement than a defined group, with widely scattered communities in varying degrees of agreement and communication.