A lot of people like Twisted. I was a huge fan for awhile, but on working with it a bit and thinking about it more I've become disenchanted. It's complex, and last I looked, a lot of it had the assumption that your program would always be able to send data leading to possible situations in which your program grows memory usage endlessly buffering data to send that isn't being picked up by the remote side or isn't being picked up fast enough.
In my opinion, it depends a lot on what kind of network programming you want to do. A lot of times you don't really care about getting stuff done while you're waiting for IO. HTTP, for example, is very request-response oriented, and if you're only talking to a single server there is little reason to need something like Twisted and plain sockets or Python's built-in HTTP libraries will work fine.
If you're writing a server of any kind, you almost certainly need to be event-driven. Twisted has a slight edge there, but it still seems overly complex to me. Bittorrent, for example, was written in Python and doesn't use Twisted at all.
Another factor favoring Twisted is that there is code for a lot of protocols already written for it. So if you want to speak an existing protocol a lot of hard work may already have been done for you.