I think there is a huge difference between American ideas about what constitutes "old money" and British ideas about "old money".
In America you need to be wealthy in order to be a member of their version of the upper class whereas in the U.K. you can be as poor as a church mouse but still be seen as old money or even landed gentry. There are plenty of impoverished aristos around.
I love the old saying "clogs to clogs" in three generations. The first generation make the money, their children spend it all and the grandchildren are left with nothing.
The problem with many nouveau riche is they don't value the principles of stewardship, something that Old Money have drilled into them from childhood.
Aristos inherit land, property, investment portfolios, jewellery, art, fine furniture etc but they don't usually view it as "theirs", they don't cash in on its value, they see it as family wealth to be preserved and increased and passed down to the next generations.
It's the same as someone planting a tree which they will never see mature or sit in its shade. When I visit stately homes and see those wonderful avenues of trees I'm always reminded that the person who planted them probably never saw them reach maturity, they knew they wouldn't but they planted them anyway. For posterity, it was part of their legacy. Legacy is important to Old Money.
A bit of a generalisation here but on balance the big difference is Old Money tends to plan for the future, husbanding resources with care whereas the nouveau riche live only in the present, spending freely but not necessarily wisely.
It's a fascinating subject. And really it all boils down to class systems. I have seen loads of these American lists about how to dress and act like "Old Money" but they don't really get it right. Especially with the teeth.....?. They concentrate on how to fake the look and maybe even the speech and certain behaviours but it goes much deeper than that. It's a whole ethos and way of being. Not something you can easily transition into.
There are plenty of real life Hyacinth Bouquets out there who get disastrously wrong. .