I've got my hands on the Amazon Fire TV Stick and put it through its paces against the Chromecast. Which is best for you? I've got specs and hands-on details.
Setup of both TV sticks is similar: Plug them into an HDMI port on your TV, go through a brief setup routine including downloading the latest software and connecting to your network, and you're ready to go. The Fire TV Stick comes with a small remote, and you may find it a bit difficult to open it up to insert its batteries. The Fire TV Stick also comes with an HDMI extender cable, which is useful if the stick itself won't fit properly into the HDMI port, for example, if you've got a device in a nearby HDMI port which blocks it. So who's the winner here? It's a tie
Ease of use
The two devices take different approaches to playing content. Chromecast relies entirely on apps or a Chrome browser on another device for playing content. To play Netflix with Chromecast, for example, tap or click the Chromecast button on your Netflix app or Web page. In fact, to play anything, you merely tap or click the button on your Chrome browser, and it casts it onto your TV.
The Fire TV Stick instead relies on a remote for accessing apps, which I find unwieldy and time consuming. To search for a movie to play, for example, you have to click out its title using the remote, which can be quite annoying. There is an Android app to control it, but there's no app yet for iOS or even for Android tablets -- it's only for Android phones. And unlike Chromecast, you can't merely cast anything on your browser to your TV -- you have to rely on the apps built into the Fire TV Stick.
The Fire TV Stick give you access to Amazon Instant Video (including Prime), Netflix, YouTube, and a number of other streaming services, while Chromecast gives you access to just about anything you can display on your browser, which makes it more flexible. You can't, however, play Amazon Instant Video using Chromecast without resorting to a workaround.
That means for most purposes, Chromecast is superior. However, if you're a Prime subscriber, or want Amazon Instant Video, the Fire TV Stick is clearly better.
So which is for you? You won't go wrong with either. If you're a Prime subscriber or care about Amazon Instant Video, the Fire TV Stick is clearly a better bet. If not, though, and you want the flexibility of casting anything from the Web onto your TV, you'll probably be happier with Chromecast.