For me, perfect pitch is a kind of memory. I knew the note was F sharp because I'd heard many F sharps before and remembered their pitch, just as I know an object's blue because I've seen many blue objects before and remember their color. I doubt if it's more mysterious than that.

Perfect pitch is useful for conductors, especially in twentieth- and twenty-first-century music, often composed without a key. The usual key-based methods of learning a score, such as pitching intervals and “hearing” harmonies in your mind, aren't always enough to help you learn contemporary pieces thoroughly or to rehearse them efficiently. If you don't have perfect pitch, you need highly developed relative pitch: when you know the pitch of one note, you should be able to pitch any other note immediately.