Get the Roku it's great for finding and adding all kids of pirated media!!!!(I kid I kid!
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Actually between the Roku and the apple device I would ask which has better aesthetics. I.e. which can you hide better in your setup or not stand out.
As per 1080P v 720P devices, for the MLB I doubt the 1080P will make a difference any time soon. Hell broadcast TV is really only 720P, given the high motion nature of sports they'll probably stay at 720Pish for a while because of bandwidth concerns.
I tried to find out what bitrate they were transmitting at and the best I can find is that for the highest quality they recommend a 3Mb/s internet connection. Now that doesn't really say they are transmitting at 3Mb/s but they might be. For 720P content at ~3Mb/s bitrate you probably won't notice much difference in visual quality between the streaming and the satellite feed.
The biggest question is in how much you use your internet for. If you use it for heavy bandwidth usages then the MLB streaming could become a quality issue if you are using heavy bandwidth somewhere else in the house. I'm betting that's probably not a concern but that's something to consider.
The only other outside issue to consider is if your internet provider has put into place monthly bandwidth caps like mobile devices. This isn't terribly common today for landline systems and most of the ones that exist are for very large bandwidth caps(again something I'd bet isn't a concern for you). But this trend will start to become more popular in the future.
Unless you're on a really small local internet provider who has pretty small bandwidth limits(i.e. pretty rare), you'd have to like be watching 100+ games a month to have this be an issue today.
You're most likely good to go once you decide a device for home.