Why do academics write/publish book reviews for journals?

In my experience, reviews are written at the request of journal editors and are typically not initiated by the person writing them. My colleagues and I usually write several book reviews every year because our supervisor is on the editorial board of the journal and tells us to write them. As reviewers, we do receive a free copy of the book, if you consider that an incentive. There is no prestige associated with writing one, and they don't count as publications in our annual performance reviews.

I wrote a book review once. For me (1), (3), (4), and the first half of (2) were all relevant factors. (Like all book reviews in mathematics of which I am aware, it was at the unsolicited request of the journal editors.)

In mathematics, book reviews are uncommon, and (contrary to cheesemeister's experience) they are somewhat prestigious to write. Only about a quarter of my book review actually reviewed the book. In the rest I described (as I was asked to by the editors) why someone would want to read a book on Subject X in the first place. I had an opportunity to "sell" my research area, and this was definitely rewarding!

Finally, it was an excellent excuse to thoroughly read the book I was reviewing, which I was eager to do in the first place. This was a lot of work, but it was very rewarding (as I could tell from the beginning that it would be). Personally, I would only review a book if I was keen on carefully reading it.

I'm a bit late to this game, but thought I'd remind you that librarians are academics! We read book reviews to help with our collections decisions. I'm a nursing librarian, so I read reviews in nursing journals regularly.