I'm going to paraphrase the eminent philosopher Chris Rock's definitions of wealth and rich.
Rich is having a lot of money. Wealth is having enough assets that you are, for all intents and purposes, unaffected by recessions and economic downturns.
It's the connotation that rich are the high earners: doctors, biglaw lawyers, professional athletes -- people that may A LOT of money, but who may have few tangible assets with which they earn more money. Wealthy people, such as the Rockafellers, Waltons, and Rothchilds, have assets in which they can overcome a financial hardship.
To me, that resonates because that means your money is financially secure. (I know I used extreme examples, but the point is that you have near-recession proof investments and assets; there are CEOs who earn 8-figures annually who live month-to-month or even week-to-week because of keeping-up-with-the-Jonesesism.)
@Jarrett Harris: That 30F30 was excellent. Probably my second favorite, after Pony Excess.