I think that for the more part, they are interchangeable. I might like to invite other native speakers to comment on this, but I have always felt that the nuance lay in what kind of 'wealth' is being spoken about. Rich: Having lots of money. You are rich if you have 2,000,000 dollars. Wealthy: Not necessarily having lots of money, but having lots of assets. You can be wealthy without being rich, because all your monetary assets take the form of houses, cars, shares etc. You wouldn't be rich if you only had 200 dollars in your back account, but you would be wealthy if you owned 2,000,000 dollars worth of assets! Affluent: This is less linked to the amount of money/wealth you have, rather this word refers to your social class or way of life. You may be rich, but if you don't live in comfort of luxury, I probably wouldn't call you affluent. Similarly, children who study in private schools often come from affluent backgrounds. In short, the distinctions are very subtle!

Rich is the general word; it may mean someone just got a lot of money (an oilman who became rich overnight) Wealthy suggests permanence, stability, someone who has always had a lot of money and who always will (a wealthy politician) Affluent usually means having a high standard of living AND some social prestige and privilege: an affluent family .... wealthy, popular, and often socially powerful. http://dictionary.reference.com/browse/rich?s=t