Rich and not famous is, by far, the better choice.
However, you have to understand that a lot of young people who want to be stars are seeking not so much wealth as fame. They don't really understand the disadvantages of losing their privacy. Instead, they're focused on the idea of being known and "loved" by millions of people, of being the center of the spotlight, of their accomplishments being celebrated, and their face up on the big screen. This kind of fantasy is particularly appealing to someone who feels they are unloved or unimportant - who seek recognition. They don't realize that fame has little to do with love, and that the public is fickle - a star is only as good as his last movie - and most of the people they meet in the industry are only interested in exploiting their fame. It's only after they have lost their privacy, and can never get it back, that they begin to realize how valuable it is.