Capability implies unrealised potential, as in "John is capable of winning", which implies John has the ability to win
but it's not definite.
It means more or less "John has the talent that's needed to win."
Ability, on the other hand, implies possibility.
"
John is able to win" means John is in the running to win but not necessarily that he has the skills or ability to win.
It means more or less "It's possible that John will win".
But here's where it gets more confusing:
"John is able to read" means John can read without doubt, whereas "John is capable of reading" means John has all the necessary brain power and whatnot to be able to read but the question of whether or not he can read at the moment is left unsaid (although the fact that someone does say "John is capable of reading" probably means John can't read right now).
english.stackexchange.com/users/90/Sarhanis
If we would like to describe the functionality of a specific product like a software, I think it is more common to use 'capabilities' (vs 'abilities'):
"MS Word has these capabilities: edit tables, copy and past, styles etc."
english.stackexchange.com/users/None/
There's another shade of meaning not yet touched upon.
able can also be of things, "of a thing (esp.
a boat):
strong, substantial, well built; or a person who is intelligent, skillful, apt, talented, or clever (OED).
Compare, "It is an able ship/
It is a capable ship.
" They are both meaningful, but I would be more likely to use the first one.
Another example, "She is an able tour guide/
she is a capable tour guide".
The second sounds contrived to me, even a little damning.
It's also interesting to note that they entered English through different sources, and that able is the older of the two.
english.stackexchange.com/users/89/Charlie
As far as I can tell, these are very nearly perfect synonyms.
english.stackexchange.com/users/78/Alan Hogue
I want to give only one condense sentence, so that you can easily to remember.
Capability often refers to extremes of ability .
I have the ability to write, but I don’t have the capability to write a novel.
Ability vs. capability vs. capacity on Grammarist
english.stackexchange.com/users/80007/Ooker
What is the difference in usage between ability and capability ?
Difference between "ability" and "capability"
english.stackexchange.com
<differences><synonyms><nouns>
english.stackexchange.com/questions/124
english.stackexchange.com/users/81/igor
There is disagreement as to whether or not these two words are true synonyms. However, it is possible that capabilities are used more to express extremes of abilities. It is also the case that functionalities of a product, for example, are more likely to be referred to as capabilities instead of abilities. "John is able to read" means John can read without doubt, whereas "John is capable of reading" means John has all theintelligence needed to be able to read but the question of whether or not he can read at the moment is left unsaid 
There is disagreement as to whether or not these two words are true synonyms. However, it is possible that capabilities are used more to express extremes of abilities. It is also the case that functionalities of a product, for example, are more likely to be referred to as capabilities instead of abilities. Capability implies unrealised potential, as in "John is capable of winning". Ability, on the other hand, implies possibility. Capability  often refers to extremes of  ability as in the example "I have the ability to write, but I don’t have the capability to write a novel. "  
These may be considered to be synonyms. However, capability  often refers to extremes of  ability. In terms of software functionality, for example, "capability" may be better. There is disagreement about whether or not these two words are exact synonyms. Some people might say that 'capability' refers to extremes of ability. When talking about software functionality, however, 'capability' is the preferred term.
"John is able to read" means John can read without doubt, whereas "John is capable of reading" means John has all theintelligence needed to be able to read but the question of whether or not he can read at the moment is left unsaid  For the functionality of a specific product like a software,  it is more common to use 'capabilities' (vs 'abilities'). These are synonyms. Capability often refers to extremes of ability . 
Capability implies unrealised potential, as in "John is capable of winning". Ability, on the other hand, implies possibility. Capability  often refers to extremes of  ability as in the example "I have the ability to write, but I don’t have the capability to write a novel. "  
There is disagreement about whether there is a signficant difference between these two words, although 'capability' may refer to extremes of ability. Capability' is generally used when discussing the functionality of software.