The difference between these words is extremely small - but usually we use able to describe current things someone can do, and capable to talk about someone's future potential. It is not a strict rule, just a general tendency.
She's able to play a song perfectly after hearing it only once.
(she can currently do this)
She's capable of becoming a successful musician.
(she has the possibility of doing this in the future)
This technology has the ability to grow crops in the desert.
(it can currently do this)
This technology has the capability to end world hunger.
(it has the possibility of doing this in the future)
The opposite of able is unable, and the opposite of capable is incapable.
The disease made him unable to move without pain.
(he currently cannot move without pain)
The treatments are incapable of curing the disease.
(there is no possibility of curing the disease with the treatments)
Note that we say able to, but capable of:
The martial artist is able to break a concrete block with his bare hands.
Good thing he's very calm - I don't think he's capable of violence.