'Capability' = one's maximum ability, to one's limits of capacity for doing/accomplishing something;
On the other hand, ability is more applicable to the generic, non-quantifiable quality of being able to do something/capable of doing something
'Ability' is closer to a talent, a natural skill which is hard to quantify, whereas capability is more applicable to enterprises, rather than persons, or, if it is applied to a person, it may more often reflect on limitations, rather than talents.
Examples:
1. Her ex-husband's capability to pay alimony does not extend to supporting her drug habit.
2. His ability to listen attracts many who want to cry on his shoulder.