How big is the difference between the sustained PhD thesis and a version publishable as a book?
It depends mostly on whether publishing your thesis as a book is a requirement in the PhD process or not. As JeffE notes in his comments, publishing the thesis as a book may be a university requirement before you get your PhD. Though, in the cases that I know, publication is after the defense, and you are allowed to make smaller modifications to the originally defended version, as long as the thesis reviewers agree to these.

Things would be different if publication is not part of the PhD requirements (maybe you have your certificates already in your hands). It's your work, and unless there are specific regulations to the contrary, I'd assume you can do whatever you like with it. In particular also publish a derived work as a book. In this case, I'd still recommend to state that it is based on your thesis (not your thesis), and discuss it with your PhD advisor.