They both mean "to think" or "to consider". Is there a usage difference between the two?
以为 carries with it the nuance of thinking or considering something mistakenly.
我以为你们已经分手了。 - I thought you had already broken up (but it seems you're still together.)
认为 doesn't have any such connotation and can be used generally to mean "to think," "to consider."
我认为他是我最重要的朋友。 - I consider him to be my most important friend.
Putting it simple:
认为 - think or consider
以为 - thought or considered (but I was wrong).