Is the difference that Kaufman and Anderson build "quirky" worlds and environments to go along with their offbeat characters? Is the problem when characters are quirky but exist in the real world, thus magnifying their quirk?
Among other things, the distorted environments do distinguish Gondry's and Anderson's films from other efforts that have been labeled "quirky," although I don't think pairing eccentric characters with an eccentric world is necessarily superior to the placement of those characters in a nominally realistic world. Often the "quirk" depends on the contrast, whether that contrast is with corporate environments, elite social worlds, or simply stubborn physics.
When someone identifies a "quirky" character, I usually expect that character to have a level of commitment to their desires that rarely involves the stealth of dissembling, much like the totally committed characters of screwball comedies in the '30s and '40s. I think if we took a classic character like J.D. Hackensacker III from
Palm Beach Story and put him in a Wes Anderson film, he wouldn't seem out of place, primarily because his clothing, his mannerisms, his voice, and his actions all signify intense personal desires/neuroses that will not bend to the environment, or at least resist the environment until a final act, whether that final act consists of a marriage or the adoption of a child.
There's a level of blindness to those characters, too, that dates back most obviously to Buster Keaton, in which a character is charmingly oblivious to alternate points of view. This partial blindness is often signified by deadpan behavior, as when a character will look stoically at some ridiculous turn of events occurring before his eyes. On one hand, he is acknowledging the event and accepting hardship, but on the other, he is looking past it toward that desire or fear that has consumed him and has come to identify him.
From what I've read, I think most people who disparagingly use the word "quirk" are leery of the revival of deadpan behavior in, most notably, Anderson's films. Perhaps it seems inauthentic to them, or perhaps confused with depressive states, like Richie and Margot Tenenbaum, or the now cliché figure of the precocious, skeptical child observing the antics of childish adults. (I wonder if vaudeville, more than method acting, produced better deadpan comics.) This mixture of classic deadpan reaction and the melancholia that haunts an Anderson film perhaps feels overstated to some. Speaking for myself, I enjoy Anderson's staging and mise-en-scène more than his characters or stories, so I can understand this critique, but
Moonrise Kingdom was the best feature film of last year that I saw, so clearly I'm drawn to these qualities, too.
I haven't read through this thread, so I apologize if I'm accidentally repeating what someone else has said.