The other big benefit to the Fire TV Stick is that it includes a remote (and an interface), whereas you're stuck controlling the Chromecast entirely through your phone. Amazon is cutting costs here, however: this isn't the same remote that ships with the Fire TV Stick, which has a microphone on it and lets you perform a voice search. If you want that remote, you'll have to buy it separately for 29.99. If you don't want to spend nearly as much money on a remote as a streaming box, then you can just download Amazon's Fire TV remote app, which also lets you perform a voice search. It's available for Android right now, but an iOS app is in the works.
It's easy to imagine that Amazon will find a lot of success with the Fire TV Stick, given the popularity of low-cost streaming devices. The Fire TV Stick is a bit more expensive than the Chromecast, but it may also be a lot more user friendly given that it presents an obvious interface for how to start streaming. And if Amazon can continue to sell these things for $19 to Prime subscribers, there won't even be a competition for many. The Fire TV is already one of the better streaming boxes out there, so if Amazon really has brought that full experience down to a $39 stick, this should be a very popular option.