"Would you do me a favor?" is close in meaning to "Please, do me a favor." "Could you do me a favor?" is actually the same as "Please, can you do me a favor?" I think the second one is just a tiny-little more polite ))

Both are correct, but with a little difference would you do me a favor? means that you ask someone if s/he doesn't mind helping you. could you do me a favor? means almost the same but it asks about the possibility, too. ............. it's my own information, so check it with someone else, too.

If asked this way, means exactly the same. Only very pedantic people would bother to make a difference. Could - past form of "can" is expressing capability Would - past form of "will" is expressing about intention "I knew that she could include some details in the title, but she would not do that" - same person, same object, same action, "could" and "would not" at the same time.