I sneaked through his base, completely undetected. I then pickpocketed him and knocked him out before leaving, and what I think that achieves (from a kind of RP perspective) is letting him know who is the greater assassin. One moment, he's doing his daily business. The next minute he wakes up to find his possessions gone. I'd like to think this makes him realize he isn't so great after all.
On my more chaotic playthrough, I simply killed him. Corvo was hungry for revenge. And he got it.