The Ottoman Sultanate became the Empire as it grew, and Caliphate refers to a specific claim to Islamic hegemony. The Caliph was considered the political successor the the Prophet Mohammed, giving the Caliphate some kind of authority (although mostly informal) over Muslims of the same denomination. For example during WW1 the Ottoman Empire attempted to style the conflict as a holy war in the hopes of turning Muslims within the British and French colonies against them.

tl;dr first 2 are just different names, 3rd is religious