They're not always the same. "Elaborate" means to give more information about something. "Specify" can have a similar meaning, but it can also mean to indicate a choice between options.

They are almost like opposites. Both mean to explain, but they do it in opposite ways. When you "elaborate" you continue to talk, perhaps on and on, to say more and more about what you want to explain. When you "specify" you hit the nail on the head, saying exactly what you mean.