There is a slight difference in meaning, but not much.
If somebody is able to do something, they can do it and I would expect them to do it.
"I am able to walk now, after the injury."
If somebody is capable of doing something, they don't do it all the time and it is not normally expected, but they could do it if they wanted to.
"He is capable of taking over the company if he gets the votes from the stockholders."
Also, capable is used to mean that somebody can not only do something, but do it well.
"He is quite capable at the piano."