Good side-by-side comparisons in this LA Times piece. You might also want to take a look at this comparison from Forbes.
I’d also keep Microsoft’s Xbox One in mind as you go through this. There are clearly a number of shared features here.
Comparing Amazon’s Fire TV with Apple TV, Google Chromecast, Roku boxes
Meh, it’s not for me. Airplay alone will keep me on my Apple TV. I already get Prime on my PS4, iPad and the TV itself. I love Amazon, but their hardware is never enough to draw me in. I’ll just keep using their services on all my other devices.
The article doesn’t mention a couple of points that I think are relevant. iOS games can be played on Apple TV via mirroring, with 3rd party controllers now appearing. With regards to the remote, I programmed my universal remote to operate my Apple TV. I use one remote for cable, TV, PS4 and ATV.