"Proverb" is certainly the most widely used of the two words. Proverbs are also sometimes simply called "sayings". Proverbs tend to be widely known and repeated, usually verbatim (sometimes with minor differences) and are often either extremely old or of unknown origin. The other notable thing about proverbs is that they tend to be advice, not necessarily "statements of truth". It is often commented that proverbs contradict each other - for example, one famous proverb says "many hands make light work", meaning the more people working on something, the better; yet another equally well-known proverb says "too many cooks spoil the broth", which means too many people working on something is a bad thing.
Looking at dictionary definitions, an "aphorism" is defined as something that contains a "general truth" or even "a concise statement of scientific principle". Many of the examples of aphorisms in dictionaries are creditable to known individuals, and from more recent history.
So, while they may be sometimes interchangeable, a proverb is less likely to be an accreditable quotation and is a piece of advice that could potentially be disagreed with; while an aphorism is more likely to be a direct quotation and be a concise statement of fact.