in theory les paroles would mean the "lyrics" you speak with mots, your sentences are constructed by putting mots together not paroles. But if you are learning the "words" of a song, you are learning the "paroles". There is also an idiom which is "donner sa parole a quelqu'un" meaning to give somebody one's word. Other than that, I would NEVER translate "words" by "paroles".
I hope this helps
The discussion here is insightful. While "mots" refers to individual words, "la parole" has a very different sense. With that in mind, a better translation of "Il apprend les paroles" would probably be "He learns the lyrics" or at least to understand "He learns the words" in the context of "lyrics" and not individual words. Check this link: http://www.larousse.com/en/dictionaries/french-english/parole/57939
I believe "apprendre" means learn if there is no indirect object, and it means to teach if there is an indirect object.
In your example: "Il nous apprend les mots", "nous" is an indirect object, so this is "he teaches us the words" or (to make the indirect object more obvious) "he teaches the words TO us"
If you take out "nous" then "il apprend les mots" has no indirect object so it becomes "He learns the words"