Wealthy has its origins in the idea of well-being and health. Rich has its origins in the idea being strong or powerful. Both refer to someone having a lot of money (or wealth or riches). If you are describing someone that has a lot of money both rich and wealthy will make the point equally well. Possibly due to its association with well-being and health people often prefer to be described as wealthy rather that rich.
- Someone that has a lot of money is often described as 'well-off'.
- Something that has a large quantity of something considered good is described as having 'a wealth' of that thing. For example: 'these books contain a wealth of information'. (But you would never refer to these things as 'wealthy')
- Food that has a lot of oil, butter or cream is described as 'rich'. (Usually as a mild criticism, as in 'it's nice but too rich for me' or 'be careful, it's very rich').