I think the Apple TV focuses on on-demand content, convenience, and usable interface, whereas the Popcorn Hour focuses mostly on compatibility with a broad range of formats.
The Apple TV features On-Demand movie rentals from every major studio, an attractive and intuitive integration with your iTunes library, a great interface to view Video Podcasts, YouTube and Flickr slideshows and it has an internal hard drive, right out of the box.
The interaction with iPods can't be understated either. You can purchase a TV show in the iTunes store, take it with you for the train ride to work or airplane, then be able to watch it on your TV when you get back. For iPhones and iPod Touches, there's also free "Remote" application you can download that lets you choose your content from your handheld device and use its virtual keyboard to search (it's very cool).
The Popcorn Hour, on the other hand, is good if you have a wide variety of file formats and you don't want to do everything the Apple way. It's more difficult to set up, and the interface isn't as smooth, but at least you won't be stuck in Apple Land.
I'd also watch the two videos to compare the user experience:
http://www.youtube.com/watch?v=BoSDo-iFpwg&NR=1
http://www.youtube.com/watch?v=uLT_T0xHzTc