Though society is a lot less hierarchical than it was in Woolf's day, there are is still a depressingly large number of people who instinctively look up to the rich and well-born simply by virtue of their wealth and exalted social status.
It is the persistence of this unfortunate attitude that makes “The Duchess and the Jeweler” relevant for our times. In the story, Woolf attempts to show us that the possession of wealth or a title is not necessarily a good thing in itself; that it doesn't necessarily bestow positive qualities on those who have them.
Take the jeweler, for instance. He may have tons of money, but it cannot buy him the class he so desperately craves. Bacon remains a vulgar, nouveau riche upstart shamelessly trying to buy his way into the aristocracy. But no matter how hard he tries, no matter how much money he spends on making his dream come true, the authentic bluebloods will never accept him as one of their own.
As for the Duchess, the possession of a title has not conferred intelligence or good judgment upon her. A hollow, foolish woman, the Duchess has lost an absolute fortune in gambling, which has led into her serious debt. In turn, this has caused her to sell off her jewelry to Bacon.
Aristocrats are supposed to be the leaders of society, people to be looked up to and emulated. But there's nothing about the Duchess' behavior that is in the least bit deserving of respect. When it comes down to it, she's as much of a phony as Bacon, which is why they are drawn towards each other. Each one has something that the other patently lacks.