I'm not sure what the difference is, practically speaking.

The difference is necessary vs. sufficient. Money and most purchasable goods aren't a significant source of happiness as best I can tell, and neither is freedom as such. Lower incomes constrain your happiness by increasing your sensitivity to negative externalities and probably also by way of status effects, but removing those constraints doesn't lead reliably to a happy life; granted, I'd expect the miserable millionaire trope to be at least partly sour grapes, but I'm sure there are plenty of independently wealthy people out there that never developed the skills to be happy. Particularly if we're talking old money, since I'd expect people who grew up with that level of privilege to respond poorly to any minor disruption. A billion dollars would give you all the freedom you need to be happy -- but going from that to "a billionaire must be unusually happy", or even "most billionaires are unusually happy", seems to depend on a lot more self-awareness and agency than I think most people of any income actually have.

I also doubt there's much of a discontinuity in the income-to-happiness curve when income reaches what Neal Stephenson called "fuck-you money"; if there was, we'd expect the lottery winner results to look different.