This is weird. I wanted to use AirPlay to send web pages and videos from my MacBook Air to my Apple TV. As soon as I did that, my network connection on the Mac failed completely. The AirPlay connection continued to work wirelessly. This was completely repeatable and reversable. AirPlay on — network down. AirPlay off — network works fine.
After a good bit of poking around in the Apple support forums I found that this only happens if… Bluetooth is enabled on the Mac. Disable Bluetooth, and maybe log out, and now AirPlay works without killing your network connectivity.
This kind of random weirdness is the exact opposite of the “It just works” that Apple fans hope for from Apple gear.