The question presupposes a differentiation between the two.
The myriad of interpretations notwithstanding, Christ, being "the anointed one" or Messiah, is an unequivocally Hebraic concept and figure.
What is called "Christian", and what is called "Jewish" are wholly different traditions and practices from the gestalt cosmology as given in the 66 books (absent Deuterocanonicals/Apocrypha). Of course that number is itself open for argument.
In the shortest possible terms: the New Testament is the 'grafting' or inclusion of the Gentiles into the Old Testament, that was exclusive to the Hebrews.
With disagreements, today's Jews refer to the Old Testament as the Tanach (Torah, Naviim, and Ketuvim - the 'law', the Prophets, and the writings) and, again with disagreement, the Tanach is subordinate to the Chumash (Torah in bound form with rabbinical commentary), being conveyed through the Talmud (Rabbinical Midrash).
Today's Christians - depending on denomination - alternate between the Old Testament as primary canon (generally Protestants), and the Old Testament as subordinate to 'oral canon' such as Apostolic Succession (Catholicism). I am unfamiliar with how Eastern Orthodoxy or Coptics view the Old Testament.
Simply put, the New is the fulfillment of the Old - the Old is the foundation of the New. They are inseparable.
The Old Testament, in its final form, serves as both a Christian canon and a Jewish canon. Despite this commonality, there exist notable distinctions in content and arrangement between the Christian Old Testament and the Hebrew Bible, also known as the Tanakh.
The Jewish canon comprises three primary sections. First, the Torah, or Law, encompasses the initial five books attributed to Moses—Genesis, Exodus, Leviticus, Numbers, and Deuteronomy. Second, the Nevi'im, or Prophets, encompass prophetic books like Joshua, Judges, Samuel (1 and 2), Kings (1 and 2), Isaiah, Jeremiah, Ezekiel, and the Twelve Minor Prophets. Lastly, the Ketuvim, or Writings, encompasses various texts such as Psalms, Proverbs, Job, Song of Solomon, Ruth, Lamentations, Ecclesiastes, Esther, Daniel, Ezra, Nehemiah, and Chronicles (1 and 2).
In contrast, the Christian Old Testament mirrors the books of the Hebrew Bible but follows a distinct organization. This Christian canon is divided into four principal sections. The Pentateuch, akin to the Torah, contains the first five books attributed to Moses. The Historical Books section comprises works like Joshua, Judges, Ruth, 1 and 2 Samuel, 1 and 2 Kings, 1 and 2 Chronicles, Ezra, Nehemiah, and Esther. Wisdom Literature, resembling the Ketuvim, features Job, Psalms, Proverbs, Ecclesiastes, and the Song of Solomon. The Prophets section is categorised into Major Prophets (Isaiah, Jeremiah, Lamentations, Ezekiel, and Daniel) and Minor Prophets (Hosea to Malachi).
Differences between the Jewish and Christian canons primarily revolve around the arrangement of books and the inclusion of additional texts in the Christian Old Testament, notably recognised by Catholic traditions as the Deuterocanonical books or the Apocrypha. These additional books include Tobit, Judith, Wisdom of Solomon, Sirach (Ecclesiasticus), Baruch, and 1 and 2 Maccabees, accepted by some Christian denominations.
In summary, while both the Jewish and Christian canons share a foundational corpus of texts, disparities in arrangement and book inclusion distinguish the Christian Old Testament from the Jewish Tanakh.