I own a Roku now and I've used my friends AppleTV a lot.
A lot of channels have captions for their shows but it's not an ALL or NONE issue. I.e. History channel has a lot of their TV shows captioned but a few of them don't have captions.
Personally I like the Roku better for the price and for the options of channels it provides.
However, BOTH the Roku and AppleTV offer pretty much the same type of channels. The only big difference is that if you own apple products, you can mirror your devices to the AppleTV but you can't do that if you have an Android. There are a lot of apps to do similar things on the Roku (for both Android and iPhone).
Caption-wise it's my experience that with AppleTV the subtitles are not synced well with Amizon Prime. They are fine with other those. I haven't experienced any issues with Roku. You can change the caption style on BOTH. However I think Netflix is locked into their yellow subtitles and there isn't anyway to change that from what I can tell.
I've also used the Chromecast and that's a boring piece of crap that's very annoying to control because you HAVE to use your phone, tablet, or computer to control anything with it and it's a pain especially when you're multitasking or if you want to give control to another person. If it had a remote, that would have been nicer. Also, the Chromecast is slooooow.