We usually mix it up because both "rich" and "wealthy" have to do with money. However, rich people try to earn as much as possible, while wealthy people think differently.
The way Paul Sullivan explained it in his article "Wealth Matters" in New York Times over a decade ago is an excellent definition of these two words. In a nutshell, Paul Sullivan's explanation goes like this:
"Wealth can be defined not as a dollar figure but in terms of what your savings allow you to do."
