The difference is that was able to fits well with contexts where the action is presented as an action that took place:
The magician was able to free himself from the straight-jacket and received a great round of applause from the audience.
When we use could with such contexts (i.e. completed action) it sounds a little odd:
The magician could free himself from the straight-jacket and received a great round of applause from the audience.
That sentence with could doesn't go well if the meaning is that he freed himself.
Slight changes present the action as one that was possible (and perhaps had been done before) but did not take place as of the temporal reference point in the past, making could viable:
The magician could free himself from a straight-jacket. [he had that skill]
The magician realized that by expelling all the air from his lungs, he could free himself from the straight-jacket. [it would be possible]
was able to also works there.
The only limitation, then, is that could does not go well when the intended meaning involves an isolated action completed in the past.
With the negative, there is no completed action, and hence no conflict for could. The reference is to an action impossible to complete in the past.
She couldn't call because she was sick.
She was unable to call because she was sick.