A proverb is a commonly recognized and punchy saying that purports to give good advice or represent a commonly accepted truth. For example, "Time and tide wait for no man". It may or may not be a quote, and in fact, in the example I gave, I don't think it is, or at least, I don't know where it came from, so I didn't use it as a quote. And that's the point of a quote: you are either explicitly or implicitly attributing it to a particular speaker or author. You are not necessarily endorsing the sentiment. For example, Adolf Hitler said, "When diplomacy ends, war begins." Sometimes, a phrase can be both. For example, "Honesty is the best policy" comes from Shakespeare.