When people hear I learned POE much faster than AnyEvent, they're usually surprised. I don't know why, but POE always made a whole lot of sense to me than AnyEvent. I've used POE in several projects (some pretty big) and it was always easy to work with, always had every component I wanted (except the SSH I wanted, though that exists too now) and the community was totally supportive and helpful. This is not unlike AnyEvent, though. I just haven't used it much.
Recently I started using AnyEvent. Not because I don't like POE anymore but because I wanted to be familiar with both. I have to say I felt quite at home with AnyEvent and started to write more and more elaborate stuff with it. I'm also trying to find more stuff to do with it, though it feels suited for more hardcore people than I. I have no idea where to start if I wanted to write an AnyEvent SSH client based on Net::OpenSSH, for instance.
If you've ever used AnyEvent, I recommend getting familiar with POE. Rocco Caputo and the community have done amazing work on it.
If you've ever used POE, I recommend getting familiar with AnyEvent. Marc Lehmann and the community have done amazing work on it.