Paramount
One of the best things about "Star Trek: Deep Space Nine" was that its space station setting meant there were more opportunities for alien characters to take on major roles. Some of the most beloved and interesting characters on the series are non-human, like Cardassian tailor/spy Garak (Andrew Robinson) and Ferengi bar owner Quark (Armin Shimerman). In fact, there are a couple of fun Ferengi folks that live on Deep Space Nine, and over the course of the series they gained some depth and the Ferengi became more than just money-obsessed villains with huge ears.
Just as Michael Dorn, who played Klingon warrior Worf, helped shape Klingon culture, Shimerman was able to influence Ferengi culture. He had originally played an evil Ferengi named Letek on "Star Trek: The Next Generation," and he hoped that by playing Quark, he could evolve the Ferengi into a more complex, complete people and wipe out past portrayals. Unlike the Ferengi of the past, Quark had empathy and was an incredibly sharp dresser, showing care in his appearance. He was also the first Ferengi to not wear a veil on the back of his head, though later the Grand Nagus (Wallace Shawn) would also be veilless.
So what gives? Why do some Ferengi wear veils and others don't? Over the years fans have joked that it was a matter of budget and that it was cheaper than creating full prosthetics. But, like the Ferengi themselves, there's a bit more there than meets the eye.
Ferengi headwear started as a quick fix
Paramount
The joke about cost-effectiveness has some basis in truth, though really the Ferengi veil was first created because of a lack of communication between two of the creative departments on "The Next Generation," as Shimerman explained on an episode of "The Delta Flyers" podcast:
"The costume design department and the makeup department hadn't conferred about how far the makeup came down and how high the costume came up. And so on 'TNG,' when I was made up as a Ferengi, when all the process was done, we realized there was a 3-inch gap of the back of my neck showing, and Michael [Westmore, make-up supervisor] went out and got a piece of material which he stapled to the rubber prosthesis."
When the Ferengi were just quick one-episode villains, there was no need to figure out the rest of the makeup. However, when Shimerman was hired to play Quark on "Deep Space Nine," he requested that they make one big change: adding earholes. In an interview with Deseret, the actor explained that the original Ferengi prosthetics required ones' ears to be pinned back, which could start to hurt after a full shooting day. "The back of the Ferengi had never been completed from the very first day that I was there playing Letek," Shimerman explained. "So since they made the earholes they decided to finish up the neck. That's the reason I don't have the veil in the back."
Quark was simply the first really important Ferengi. So, we have a behind-the-scenes explanation as to why some Ferengi wear veils and others don't, but what about the in-universe reasons?
The canon of Ferengi veils
Paramount
Over the course of the dozen or so amazing Ferengi episodes of "Deep Space Nine," some Ferengi wear veils while others do not. Notably, Quark's brother Rom (Max Grodénchik) and his son Nog (Aron Eisenberg) both wear veils in almost every episode save for "Take Me Out to the Holosuite," in which they wear baseball caps. Nog even has a veil in his Starfleet uniform, which means they have some cultural or religious significance in order to be allowed. Starfleet isn't even ok with Bajoran earrings, so clearly the veils have to mean something. Unfortunately, there hasn't been a definitive canon explanation and fans have mostly been left to guess. There is one tiny clue, however, in a correction from the L.A. Times where Shimerman wrote in as Quark to correct where the paper had posted a photo of Nog with his name. In the letter, "Quark" chides the paper, asking, "Do all we Ferengi look the same to you? You should be ashamed! Not only is there an obvious height difference, but I haven't worn a veil in the back of my head for ages."
While this was probably just Shimerman having a bit of fun, it speaks to the idea that Ferengi veils show some kind of lower status (which Quark would have abandoned once he became a business owner, and the Grand Nagus would certainly not have). It's funny that Nog and Rom both wear veils, which might be seen as "traditional," when they are two of the biggest tradition-breakers of all. After all, Nog joined Starfleet and Rom eventually ushers the Ferengi into joining the Federation when he becomes Grand Nagus! Whatever the in-universe reason might be for the veils, one thing is for certain: Rom and Nog make them look good.