Some of the most obvious differences between the Chumash and the Quran is the language, style and organization. Where the Chumash is written in Hebrew, Biblical Hebrew more correct, the Quran is written in Arabic. Where the Chumash is more or less told as long story, presented chronically, with Deuteronomy somehow standing out from the four other books, the Quran seems to be a mix of random revelations, dealing with various themes. Where the Chumash is beginning from one end, with the creation of the world, and ending at the other, with the death of moses, the Quran is organized after the length of the chapters, the suwar, with the longest first and the shortest last, except the opening surah, surat al-Fatihah. We are not told which suwar are from Mecca or which are from Medina, though the general notion is that the short suwar are Meccan, while the longest are Medinian.
The Chumash seems to want to tell a story, where the Quran is more focused on explaining various conditions appearing during the life of Muhammad. True, there are parts which relates to earlier prophets, but the appearance of these parts seem to be provoked by either incidents needing them or questions about them. See for example when Muhammad reminds the Children of Israel of Allah’s former favors bestowed upon them (2:40 and 2:47), or relating to Abraham (2:124 and 6:161). The Quran is a constant dialogue involving its readers and reciters. The Chumash on the other hand relates a story, telling about what happened to the pre-Israelite world (in Genesis) and the Israelites themselves (Exodus and onwards). Of course, when the religious Jew is studying the Chumash, he – as much as the Israelites being told about – takes part in the incidents. He is not outside, but inside the Biblical account. He too was present when the Israelite received the Torah at Mount Sinai. But this is the traditional way of relation to and studying the Chumash, based on interpretations of it.
The relation to the languages of the two Scriptures is only explained in the case of the Quran. According to 12:2 the Quran was revealed in Arabic, in order that “you,” the Arab tribes, would understand it, and this is being expanded all through the Quran. The awareness that the Quran is being revealed in Arabic is very central, which can be seen from the many places this is being mentioned, whether when it is outright stated that the Quran is in a “clear Arabic language” in order to make it “easy” to understand(16:103, 19:97, 26:195, and 43:3). But that is not the whole purpose of the Quran being in Arabic, it is also in Arabic in order to be a warning ( That there are non-Arabs is also considered by the Quran (20:113 and 42:7) whether it is to warn the individual or the “Mother of Cities” (Mecca). There are other verses dealing with Arabic as the language of the Quran, but this is enough to show how central the awareness of the Quran being an Arabic revelation is. We don’t see the same focus on language in the Chumash, only relating the language in relation to the tower of Babylon, where it states that “all the nations were of one language,” and how God changes this in order to confuse them. The Hebrew language of the Chumash is not explained, except – maybe – in relation to Abraham and his descendants being descendants of a Hebrew, themselves Hebrews, and in that regard simply taking it for granted that their Holy Scripture is in Hebrew as well. But still, if we relate to the wider context of the whole Jewish Bible, we don’t see anything of the same awareness of the language, even having some books in another language, e.g. the Book of Daniel and the Book of Ester, both being in Aramaic. It seems that Aramaic in later times was as much the language of the Hebrews as Hebrew was.
Though the two bodies of writings might seem very different in their structure, where they have things in common is their followers reverence for them. Both the Chumash and the Quran take the central focus par excellence in Judaism and Islam. Both are found as the basis for any legal decision or any discussion on metaphysic matters. Both takes the focus as the main object of study, whether it being the Jewish tradition of reading the whole Chumash during a year or the Islamic ditto with the Quran during the holy month of Ramadan. We see it as well in the discussions in the Talmud, which mostly are related to and centered on Biblical verses, such as the discussion of the three daily prayers, which are related to the practice of the three patriarchs, Abraham, Isaac, and Jacob.
If we relate to the various sectary strives in both Islam and Judaism, we see the same centrality and acceptance of the Quran and the Chumash, whether it be the Sunni-Shi’a conflict, or the various Jewish groups either accepting or refusing the Oral Tradition, all Muslim groups accept the central status of the Quran, as well as all Jewish groups to our days have accepted the status of the Chumash. This is so central, that it might even be possible to deem a sectarian group either outside the Islamic sphere or the Jewish sphere, religious speaking, in their relation to the Quran and Chumash respectively.
So while we do find many differences between the Quran and the Chumash, that is when we relate to the two Scriptures as pure texts, not as holy religious scriptures. In that matter the reverence shown them by their followers, shows a very similar attitude.