Is the Christian old testament comprised of exactly the same books that are in the Jewish Bible (I think labelled Tanakh)? Are there any differences between the Christian old testament and the Jewish Bible?
The Christian Old Testament is the Jewish Bible.
The Old Testament is a Christian book. It's based on the Jewish bible with often the same books (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 to the Hebrew Bible that Judaism uses.
The term Old Testament was invented by an early Christian (I'm not recalling his name), who's goal was to say that Judaism was defunct and their texts were "old" and "superceeded." It's part of the whole anti-Judaism theme that ran through parts of Christianity's implemntation over the years. So from a Jewish perspective it's considered very negative to refer to their texts as "OT."
Also in Judaism it's not considered testament. In Christianity that concept of giving testament to the truth. In Judaism it doesn't make sense to Judaism's themes & purpose.