A rant: The Bingleys rank higher socially than the Bennets. Yes, their fortune is from trade while the Bennets are landed gentry, but trade wasn't an absolute social stain in Regency England. Indeed, Sarah Sophia, the Countess of Jersey and one of the Lady Patronesses of Almanacks, was also an active banker.

Social status clearly wasn't a one-issue thing, multiple things went into it. Here for example are Emma's thoughts on Mr Elton daring to propose to her:

he must know that in fortune and consequence she was greatly his superior. He must know that the Woodhouses had been settled for several generations at Hartfield, the younger branch of a very ancient family—and that the Eltons were nobody. The landed property of Hartfield certalinly was inconsiderable, being but a sort of notch in the Donwell Abbey estate, to which all the rest of Highbury belonged; but their fortune, from other sources, was such as to make them scarcely secondary to Donwell Abbey itself, in every other kind of consequence; and the Woodhouses had long held a high place in the consideration of the neighbourhood which Mr. Elton had first entered not two years ago, to make his way as he could, without any alliances but in trade, or any thing to recommend him to notice but his situation and his civility.

So here we have family history, fortune, civility, and the views of the neighbourhood as important considerations, as well as landed fortune. How do the Bennets and Bingleys compare on these?

Mr Bingley's income is over double that of Mr Bennet's and his sisters have large dowries. JA tells us they were:

...in the habit of ... associating with people of rank;

In support of this, they socialise with the Darcys and Louisa is actually married to a man of fashion (more so than of fortune :) ) who has a London house.

JA also tells us:

They were of a respectable family in the north of England..

And "respectable" could mean very high status, Lady Catherine, in her famous confrontation with Elizabeth, describes the Darcy's and de Bourgh’s as:

My daughter and my nephew are formed for each other. They are descended, on the maternal side, from the same noble line; and, on the father’s, from respectable, honourable, and ancient—though untitled—families. [Emphasis mine]

Given that the family connection was impressed on Caroline and Louisa's minds, I suspect the Bingley family were fairly high status. Personally I think the Bingley's grandfather might have been a landed gentleman who married the well-dowried daughter of a merchant, and their father was a younger son who got started in business with the help of his mother's connections.

As for the views of the neighbourhood, everyone is eager to call on Mr Bingley when he arrives, and JA tells us that Bingley is generally likeable:

Bingley was sure of being liked wherever he appeared...

Could one say the same of Mr or Mrs Bennet? Indeed the behaviour of the Bennets, with the exceptions of Jane and Elizabeth, lowers their status.

So the Bingleys are much richer and better connected than the Bennets, and Mr Bingley has the edge in civility and behaviour.

We see their high status play out in the novel: that Darcy thinks the Bennets' status is so low that at first he can't possibly be falling for Elizabeth, and he acts to separate Bingley from Jane is the main driver of the plot. Conversely JA tells us Darcy wished for Bingley and Georgiana to marry. It's not just Darcy either, Caroline and Louisa object to the match even though they like Jane socially, and when Bingley finally gets engaged to Jane their neighbours think she's the lucky one:

The Bennets were speedily pronounced to be the luckiest family in the world...

Even Lady Catherine says to Elizabeth that:

I was told that not only your sister was on the point of being most advantageously married,...

TL;DR: one could be in trade and of high social status, the Bingleys have other markers of status, and everyone in the novel treats them as of higher status than the Bennets.