Perfect and simple use.

If a speaker has looked for a book on a shelf and the book is not there, can he say both of these to a friend?

"Have you put the book back on the shelf?"

"Did you put the book back on the shelf?"

If he can't use both...why can't he?

"Have you put the book back on the shelf?" I think this one is more used by the British English speakers.

"Did you put the book back on the shelf?" and that one by the American English speakers. Correct me if I'm wrong!

I think, in that context, you can't say the first one in either AE or BE.