oboerareruyou ni and oboerukoto ga dekiru you ni have no "meaning" difference. You are literally saying the same thing. But it is far more natural to use the potential form plus you ni when describing wanting to be able to do something as a reason for why you are doing some other thing.
"tomodachi to isshou ni asoberareru you ni oyogu no renshuu shitemasu." is more fluid and natural sounding than "asoberu koto ga dekiru you ni" in that sentence. Think of it like the difference between saying: "so that I will be able to play" vs the more natural: "so I can play"
"ga dekiru" on the other hand, is most often used on its own, to make a special point of explaining something that you can do.
watashi wa piano hiku koto ga dekiru. As a simple declarative of ability. This is by far the most common usage of the "ga dekiru" pattern.
I have heard "ga dekiru you ni .... etc etc." before, but it's rare, and as far as I can tell it's used mostly to strongly emphasize striving for the ABILITY to do something, where that ability is the main goal... the difference between "studying japanese so I can understand it" and "studying japanese so I'll be able to understand my wife's parents"