Ability means the physical, mental, financial, or legal power to do something (ability to walk, to read, to pay a bill, to assess property). Capacity is the ability to hold, absorb, or contain (a bucket with a capacity of 1 gallon, a motel filled to capacity). Ability takes varied prepositions (ability in physics, ability with languages, ability at riding horses). Ability is often followed by an infinitive (ability to think clearly, not of thinking clearly). Capacity is followed by the preposition for (capacity for hard work) or of (capacity of 10 quarts).
Ability is a more positive quality or attainment than is capacity. A person may or may not have the capacity to learn singing; after study and practice he may or may not have the ability to sing. Approximate synonyms for ability are competence, skill, aptitude, faculty, and talent.
Dictionary of problem words and expressions. Harry Shaw. 1975.