Starting with Quark and the launch of DS9, Ferengi characters started showing up sans headdress in both DS9 and TNG. Other than the noted exceptions of Rom and Nog, it seems only speaking Ferengi do not wear headdresses. This is probably to cover up prosthesis seams easier than making a full upper torso piece.
As with many canon uniform/outfit questions we can only speculate since it is not directly mentioned in the shows and may be an artifact of current fashion without greater symbolism. But looking at the characters who wear or don't wear the headdress we can infer their symbolism and meaning. All Ferengi that we encounter without the headdress are all known as businessmen, are successful financially, or otherwise dictating their own path (in the case of Dr. Reyga). And those who wear them are either adolescents or those who are not entrepreneurs.
Also, headdresses seem to be somehow related to the earlobe, usually continuing from the top of one lobe to the other. This could be related to the cultural superstition of earlobe's relationship to business acumen, as seen in the of repeated phrase "the lobes for business".
Possibly they originally were created as some sort of hat (considering their home climate) that emphasized the earlobes because of the superstition tying them to success. But as time went on it was considered a sign of small lobes to wear the hat, so it eventually shrunk to the tasteful back band headdress. Quark, Brunt, Zek, and others don't wear them because they are confident in their lobes and don't need any adornment to emphasize them. A parallel could be drawn to the human top hat or fedora, originally used to indicate status, but has fallen out of favor and seen as garish or humorous.
And Moogie, well she is female, no headdress for her. :)
Notable examples of Ferengi along with their headdresses:
- TNG s1e05 - All three Ferengi wear headdresses
- TNG s1e09 - DaiMon Bok and crew wear headdresses
- TNG s3e08 - All Ferengi wear headdresses
- TNG s3e24 - All Ferengi wear headdresses
- TNG s6e07 - Ferengi pirates all wear headdresses
- DS9 s1e01 - Quark does not wear a headdress, Nog does, as do Ferengi bartenders, this seems to be standard going forward
- TNG s6e22 - Dr. Reyga, does not wear one
- DS9 s1e11 - Rom wears a headdress (will continue for the whole series), Nagus Zek does not, Nor does Gral and Nava (both pointed out for their business acumen and had speaking parts)
- TNG s7e9 - DaiMon Prak does not wear one
- DS9 s2e07 - Pel, independent business "man" does not
- TNG s7e22 - Bok (former DaiMon) still wears one
- DS9 s3e23 - Brunt does not wear a headdress
- DS9 s4e16 - All Ferengi workers of Quark's bar wear headdresses.
- DS9 s6e10 - Galia (businessman) does not wear a headdress, Leck (assassin) does
- DS9 s6e23 - Nilva (important businessman) does not wear a headdress