There's also implications of social class in "Wealthy" that aren't as common with "Rich".
In other words:
If you say that Bill Gates is rich, you are saying he has lots of money.
If you say that Bill Gates is wealthy, you are saying that he is rich, & that he grew up with lots of money.
A thesaurus at YourDictionary.com explains the difference as:
rich is the general word for one who has more money, possessions, or income-producing property than is necessary to satisfy normal needs; wealthy adds to this connotation of grand living, influence in the community, a tradition of richness, etc.
William Safire, in a 1990 column, puts it as, "a rich family has to wait until its money ages before it becomes a wealthy family."
