What is the difference between 认为 and 以为?
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.