And there is also a certain asymmetry, because popularity at least can be bought to some degree but converting pure fame into lasting wealth is hard work if, depending on the circumstances, possible at all.
 My preferred definition is actually the ability to do what you want, when you want, and with whom you want, which is a little broader than simply money.
That's because it used to be true and was for quite a while. Fame means lots of people know who you are, which requires widely amplifying and broadcasting your identity and work. For most of modern human history, technilogical limitations made that broadcasting very expensive. You had to spend some money for each reached human. Think printing pamphlets in Dickens' era.
This meant that, generally, only the rich could afford to become famous. Fame following money and the direction was very rarely reversed aside from occasional cases of infamy like mass murderers.
Broadcast TV made that much cheaper. A single show could reach millions. But production was very expensive so even though the marginal cost per viewer was low, the barrier to entry was still very high. That meant few got in and those were mostly otherwise well connected or part of an established privileged class.
You do start to see an increasing number of "marginally famous" people here who got recognition from being guests or contestants on shows. Think "Jerry Springer" famous. These people tend to be quickly forgotten but have the misfortune of experiencing everything about fame with almost none of the money.
Then the Internet and video streaming happened. Now the barrier of entry is virtually zero — everyone has a smart phone that can shoot video. The marginal cost is zero — ads pay for distribution so the producer fronts nothing. Some money comes in, but its very little. So now there is a larger and larger group of people for whom fame came first and wealth came later or never.
I don't think our culture has caught up to that reality yet. There's still a presumption that anyone famous always has enough money to deal with the downsides but that's sadly not true. I honestly feel bad for people like mid-level YouTubers who have stalkers and death threats but are effectively making minimum wage.