It depends on what you want in life and how good you are at dealing with the problems that come with fame.
I have been both, even though I was not particularly rich at the time I was famous. I didn’t mind the media much, since I had nothing to hide, but the worst bit is that the phone was ringing off the hook at all sorts of times. Journalists would call at 4am because they had a deadline, or at 10am when I’d still be sleeping, assuming 10am is late enough.
Other famous people I have met seem to have a much normal life than you’d expect. Especially if they’re famous for something like writing a book. Sure, there are the Pamela Andersons too, but I think they may be sort of looking for trouble themselves. Media interest keeps them famous, and even being notorious is better than being forgotten. (more or less what april just said)
Overall, I was not very happy with my “famous” spell, especially since it did not leave me much time to do the thing I was famous for – music. As a matter of fact, my second album is pretty crap, and I think that could be part of the reason.
I have an actress friend who spends so much time doing interviews and going to cocktail parties that she doesn’t have any personal life, and doesn’t really make all that much when you think about it.
I learnt my lesson back in 1996 and when I started over, I made sure to concentrate on the singing and generally keep a low profile. I’d go into a city full of my posters incognito, set up the stage, sing my songs and get out of there the same night. No interviews, no groupies, nothing. Maybe the odd autograph right after the show and that’s it. I made a lot more money that way and had a much healthier and balanced life. And always slept at home, even if it meant getting there the next morning.