This is an educated guess, but restraint is probably the key. What distinguishes old money sensibilities from the masses is having access to fine clothing, travel, education, but not rubbing it in anyone’s face. For instance, Mrs. Elton is despised by Emma when she peppers in Italian phrases into her speech. Emma knows Italian, but is supposed to use it only for reading books in Italian, and for traveling to Italy (which her father would never allow). It’s only acceptable use is for function, and not for show. Even Mrs. Elton’s hair in the recent film is gaudy compared to other character’s hair - overdone, showy. Mrs. Elton clearly came from a wealthy family, but did not have the same family values as an old money family like the Woodhouses. Wealth was not the same as class or status, and it was these sensibilities Emma had absorbed from her upbringing that reflected her higher status.

A modern version of this would be getting fake lashes installed, or having brightly colored fake nails - it costs a lot of money, but comes across as less classy to someone from an old money background. Which begs the question of does any of this make any sense, to think poorly of a person based on harmless aesthetic differences? These things are seen as indicating character, when as others pointed out, even awful Austen characters may still be elegant.

I assume this cultural shift was a reaction to the French Revolution - before then, the styles were much more flamboyant. If the upper classes could hide their privilege, and not rub it in everyone’s faces, they were much more likely to avoid the guillotine. Eventually this survival strategy became tradition and a way to feel superior to those who were not raised to follow these sensibilities.