Well there's the obvious difference that Christian Bibles have the New Testament, which is entirely early Christian literature and thus not relevant to the Tanakh. Most Protestant Christian groups use an Old Testament that is identical in content to the Tanakh but ordered/organized differently.

Both places the Torah, the Ta- in Tanakh, first. However, Christian bibles typically follow that with books of history, then books of poetry, and then books of prophecy. The -ka- in Tanakh denotes the Nevi'im, meaning Prophets, the -kh for Khetuvim, meaning writings. Notably, some of the "histories" and "prophets" are in different corresponding categories. There's also some minor differences in that Christians tend to split books in two that Jews group together (mostly those marked as 1 & 2 in Christian bibles)

However, that isn't the only Christian canon. The Roman Catholic Church recognizes several books, or sections of books, of pre-Christian Jewish literature not included in the Tanakh, like 1 & 2 Maccabees. These are usually described as the Apocrpha. Orthodox churches typically include all that and a few more books in their canon, and the Ethiopian Tewahedo Church famously incorporates numerous otherwise unique books into both Testaments.