Hebrew Bible is a term that refers to the common portions of the Jewish and Christian biblical canons. In its Latin form, Biblia Hebraica, it traditionally serves as a title for printed editions of the masoretic text.
Many scholars advocate use of the term Hebrew Bible when discussing these books in academic writing, as a neutral substitute to terms with religious connotations. Additional difficulties include:
Hebrew in the term Hebrew Bible refers to the original language of the books, but it may also be taken as referring to the Hebrew people, who originally wrote the books, and the Jews of the Diaspora, who guarded transmission of the text up to the age of printing.

Usage of Old Testament does not refer to a universally agreed canon, but rather to different sets of books depending on denomination. Christian use of Old Testament is occasionally misunderstood to allude anachronistically to covenant theology or dispensationalism (these terms post-date Old Testament by more than a millennium). It can also be misunderstood to allude to a lesser known, and even more recently coined term, supersessionism. Any such allusions are naturally inimical to confessional Jewish writers.
Tanakh is a Hebrew acronym unlikely to be appreciated by readers unfamiliar with that language. It also refers to the particular arrangement of the biblical books as found in Judaism, and to the specific textual variants found in the masoretic text. This arrangement and the textual variants are not part of the Christian canon. Confessional term
Another important issue relevant to use of Hebrew Bible rather than Old Testament is the documented misunderstanding of the sense of old in Old Testament. In Christianity old in Old Testament essentially refers to time. In French it is fr:Ancien Testament, in Latin Vetus Testamentum (like Vetus Latina Old Latin), in Greek hē palaia diathēkē (Ἡ Παλαιὰ Διαθήκη, palaios gives several English prefixes like palaeography). There is additional, confessional implication, but the semantics of this is non-trivial.
Christian commentary on the New Testament understanding of the relationship between the Testaments became controversial in the 2nd century. Consensus was eventually achieved, well before the Catholic-Orthodox division, so all major divisions of Christianity have inherited that consensus.
The controversy arose when Marcion and his followers held the Hebrew scriptures to be superceded. So strong were Marcion's views that even New Testament books that quoted the Old were excluded from his canon. He was not entirely consistent in applying this rule, because nearly every book of the New Testament makes such quotations. Along with Gnosticism, this view has the dubious distinction of being one of the first to be classed as heretical by the early Christian "peer review" process. Under this interpretation, old in Old Testament refers to the age before expansion of the covenant through the Messiah.
The New Testament documents themselves present Jesus and his followers as being opposed for preaching this message of gentile (non-Jewish) inclusion. Essentially, the New Testament appropriates the Jewish tradition the benefit of Christians. This is a serious matter for believers in both faiths, and a matter that scholars of those faiths often wish to leave out of contention when co-operating on projects of common interest, such as the Dead Sea Scrolls. This is another reason non-confessional terms like Hebrew Bible suit themselves to academic, and other, discourse.

Specific canons

Books of the Bible for the differences between Bible versions of different groups, or the much more detailed Biblical canon.
Greek Scriptures
Masoretic Text, the standard Hebrew text recognized by most Judeo-Christian groups.
Society of Biblical Literature, creators of the SBL Handbook which recommends standard biblical terminology.
Table of books of Judeo-Christian Scripture
Non-canonical books referenced in the Bible
Torah