All these overlap with one another, sometimes a little, sometimes a lot, sometimes completely. An aphorism is a short statement expressing a piece of wisdom, often wittily put. An adage is an old saying that has become accepted as a truth. An apothegm (pronounced AP-uh-THEM and also spelled apophthegm) is a terse, witty saying. A maxim is a concise statement of a principle, a truth, or a rule of conduct. A proverb is a short pithy saying, expressing a folksy truth that is widely accepted. An (old) saw is a familiar saying, much worn and sometimes distorted through long use. And a saying is almost any of these. See ADAGE; SAW (2).
