Rich is having lots of money. Wealthy is having a lot of passive income. In practice, the 2 are similar. A rich person can invest, and so become wealthy. A wealthy person is often rich. It's possible to only be 1 however. E.g. Someone with a large property portfolio would be wealthy, but not necessarily rich.
This is one of the powers of the English language. We have a ridiculous number of pseudonyms. They have similar meanings, but vary subtly. This allows us to express ideas with a lot more granularity than a lot of other languages.