Does the Christian Old Testament include exactly the same books that are in the Jewish Bible (called the Tanakh in Judaism)? Are there any differences between the Christian Old Testament and the Jewish Bible?
The term "testament" can also be translated as "pact" or "covenant". The notion of a coming new covenant is a completely Jewish one, as it first appears in the prophets (e.g. Isaiah 59.21; Jeremiah 31.31, 32.40; Ezekiel 37.26). Jesus, the Messiah, established this new covenant through his death at his last Passover with his disciples (Mark 14.24; Luke 22.20; Hebrews 8.8, 10.16-17). Thus, the term "New Testament" refers to the scriptural writings inspired by God under this new covenant. Those Jews who reject Jesus as the Messiah obviously do not believe that a new covenant has been established and so reject the validity of the New Testament as well as the term "Old Testament" for the Tanakh.
This link offers a good explanation of the difference between the Jewish Tanakh and the Christian Old Testament in terms of the order of the books, which are the same for both. The author prefers the order in the latter based on its symmetry, but argues that both were probably in existence during the first century, when the discussions and arguments were taking place between Jesus (and subsequently, his apostles) and the Jewish religious leaders.
The Old Testament is based on the Jewish Bible with often the same books. (The Catholic Church uses variations in books compared to the Protestant groups, and it all overlaps with but isn't identical to the Jewish inclusions.)
However more importantly, the translations for Christian versions (aka Old Testament) include translations that would not be considered accurate in Judaism. Therefore while one can see the same title and say it's the same book, it's not entirely the same book in content as it comes over into English.
The term Old Testament was invented by an early Christian (I'm not recalling his name), whose goal was to say that Judaism was defunct and their texts were "old" and "superseded." It's part of the whole anti-Judaism theme that ran through parts of Christianity's implementation over the years. (So from a Jewish perspective it's considered very negative to refer to their texts as "the Old Testament.") This is why some of those differences in translation exist, and why it's not even intended to be "the same" books entirely.