The Fire TV runs Amazon's modified version of Android, giving it access to all the apps and games in the retailer's admittedly modest Appstore. It has streaming support for all of Amazon's video and music content. There's also third-party support for Netflix, Hulu, and plenty of other stuff.
It's still more expensive than the Fire TV Stick or a Chromecast, but it does a lot more. You also get Amazon's voice-enabled remote, which doesn't come with the Fire Stick (it has a cheaper non-voice one). As usual, this is a US-only deal.