Hi dmp: one is a verb and one is a noun, so look to see whether "poder" is used as a subject or object in the sentence - then it is 'power'. If it used as the verb - then it is "able to" or "can" - however it will often be conjugated to match the subject (puede, puedes, podemos, etc)
The verb poder always indicates the power to do something; so by extension, to be able. It is only when translated into English that it possibly becomes "able". The original word in Spanish remains poder (power). When someone says "Puedo ir" they are not thinking "I can go" or "I am able to go", the closest English translation is "I have the power to go". This is consistent then with the noun meaning of poder, which is also (and only) power. So what you are trying to argue is that the possible but limiting translation of the verb poder to "able" should also allow the meaning of the noun poder to map to the noun of "able", "ability". That is a bad habit to get into.
As for the noun "ability", there are several Spanish words that would be used if that was the intention of this phrase, although only habilidad is at all likely.