What's the difference between an "aphorism" and a "maxim"?

An aphorism is a synonym for a proverb, that's also short and to the point. Usually, it's supposed to impart some important moral lesson. Here's a well-known example:

A bird in the hand is worth two in the bush.

A maxim is a slogan or a personal prescription for some generality of life; importantly, it does not have the same moral connotations as an aphorism. Maxims can be downright dangerous; consider

Four legs good, two legs baaaddd.

from the famous Orwell satire Animal Farm, a phrase famous for connoting the kind of totalitarian mindset required for enforcing the simplistic sociology Orwell found in communism.

A big difference between the two words is that aphorisms need be drawn from society somehow -- they are known to people other than yourself, they are received wisdom. In contrast, one can have personal maxims; it's fairly common nowadays for people to tattoo such on their bodies, things like "Faith and Family" in elaborate script to apparently remind them how to live.