There is more behind Lady Catherine de Bourgh's sense of superiority to the Bennets than simply the fact that her nephew is so much richer than they.
In the conversation you quote, after Elizabeth points out that both Darcy and Mr Bennet are gentlemen (i.e., owners of inherited landed property), Lady Catherine de Bourgh retorts:
“True. You are a gentleman’s daughter. But what was your mother? Who are your uncles and aunts? Do not imagine me ignorant of their condition.”
We know of Mrs Bennet's family background from Chapter 7:
Her father had been an attorney in Meryton, and had left her four thousand pounds.
She had a sister married to a Mr Philips, who had been a clerk to her father and succeeded him in the business, and a brother settled in London in a respectable line of trade.
An attorney might seem like a fine, upstanding, gentlemanly profession. However, in Regency England, it meant that you had to work for a living, and to make your own way in the world. That is, you were not born into landed property—you were not gentry. As for trade, it was again a respectable way to earn one's living, but considered a bit sordid nevertheless. You were supposed to inherit money, not make it, and tradesmen spent their entire time thinking about making money. So these livelihoods, simply because they were livelihoods, marked their practitioners as less than quality. The "quality" simply inherited landed property, or had ancestors who, even if tradesmen, had made and invested enough money that their descendants did not have to work for a living.
So yes, Mr Darcy is not titled. But his aristocratic background and obscene wealth makes up for this. Neither of these factors pertains to Mr Bennet. And if Darcy is connected upwards to the aristocracy through his family connections, Mr Bennet is connected downwards, to clerks and tradesmen. This makes Elizabeth's family an unequal match for Mr Darcy's.
Bingley's sisters are at pains to point this out to both Darcy and Bingley, for fear of the men's attractions to Elizabeth and Jane respectively. In Chapter 8, Louisa Hurst is speaking:
“I have an excessive regard for Jane Bennet,—she is really a very sweet girl,—and I wish with all my heart she were well settled. But with such a father and mother, and such low connections, I am afraid there is no chance of it.”
“I think I have heard you say that their uncle is an attorney in Meryton?
“Yes; and they have another, who lives somewhere near Cheapside.”
“That is capital,” added her sister; and they both laughed heartily.
“If they had uncles enough to fill all Cheapside,” cried Bingley, “it would not make them one jot less agreeable.”
“But it must very materially lessen their chance of marrying men of any consideration in the world,” replied Darcy.
To this speech Bingley made no answer; but his sisters gave it their hearty assent, and indulged their mirth for some time at the expense of their dear friend’s vulgar relations.
This is of course self-serving. Bingley's sisters want Bingley to marry Georgina Darcy, and Darcy to marry Caroline. But while the Bingleys have enough money to be able to live off the interest on their investments, and therefore meet one criterion for "quality," they are not as far removed from trade as Louisa Hurst and Caroline Bingley would have one think. Mr Bingley does not own landed property, and neither did his father. Their inherited wealth is from trade. From Chapter 4:
Mr. Bingley inherited property to the amount of nearly a hundred thousand pounds from his father, who had intended to purchase an estate, but did not live to do it.
In other words, their money is new money. They just have enough of it for its less than impeccable antecedents to be overlooked. But a large part of Caroline and Louisa's motivations for the matches they have in mind is that these matches would enable them to transcend their arriviste status. Marrying into the well-connected and fabulously rich Darcy family would consolidate the Bingleys' ongoing climb up the social ladder and enable them to escape the taint of their low origins. See also this question and answers thereto for a discussion of similar dynamics in Austen's Emma.
By definition, the gentry was land-owning classes one step below the aristocracy. So Mr Bingley is not, in fact, gentry, while Mr Bennet, who does own land (albeit entailed), is. In the question, when Mr Darcy, Mr Bingley, and Lady Catherine de Bourgh are all lumped together, and Mr Bennet is asked about in opposition to them, it is worth pointing out that all four occupy rather different places in the peculiar British pecking order. Lady Catherine is entitled (as alexg points out in their answer); Mr Darcy is a very wealthy landowner, with aristocratic relatives and no worries about pesky entails; Mr Bingley is just one generation removed from "vulgar relations"; and Mr Bennet, alas, is married to a woman who is vulgar in as many senses as Lady Catherine is entitled.