能う is translated as being able to do something or having capability to do something. How is its meaning and usage different from できる and potential form of a verb?
能う is a word mainly used in classical Japanese and kanbun. Many native speakers who are not particularly interested in literature grow up without even knowing this word until they start to learn classical Japanese at high school.
The meaning of 能う is "to be able", but this was almost always used in its negative form, 能わず, and meant "not to be able." Examples are found here, although most of them are classical sentences. ～こと能わず is the most common form which can appear in stiff fantasy novels and such. For example, 見ること能わず roughly means 見えない.
Unless you are a creative writer who needs to simulate old writings, you should not be using this word regularly.
能う is an old word. So, it is not common to use this word different from using できる commonly.
And goo dictionary says
１ 可能の意を表す。なしうる。できる。「行くこと―・わず」「味については議論するに―・わず」 「危きに臨めば平常なし―・わざる所のものを為し―・う」〈漱石・吾輩は猫である〉
this means "being able to do something or having the capability to do something".
２ 理にかなう。納得がいく。 「―・はぬことなり。はや出だし奉れ」〈竹取〉
this means "it makes sense".
３ 適する。相当する。 「此れ汝が着る物に―・はず」〈今昔・二三・一八〉
this means "it is appropriate".
Hope it helps.
According to RikaiKun app for Chrome, できる is to be up to the task of doing something, or to be ready or capable, whereas 能う is to be skilled at it.