to me, resolve gives the impression of a compromise. like jacinta said, if you have two differing opinions, and you resolve the argument, that could mean that the two people compromised rather than one or the other winning. There could be many ways to resolve an argument.
so to resolve something you need to have some sort of conflicting options in the first place.
interestingly, I wouldn't tend to use "resolution" in this situation. If my friend and I resolved our argument, I would still say that we found a solution to the argument! I don't know why, maybe because these days "resolution" is used more as a computer term?
anyway, that's just my opinion!