[Inziladun]

Gordis

Frodo's clothing disappears with him because he wears the Ring. Nazgul clothing remains visible because the nazgul do not wear their Rings - Sauron keeps the Nine himself. When the nazgul desire to go around invisible, they have to remove the clothing (as in UT-Hunt for the Ring), because Sauron doesn't deign to lent them their Rings.

Not to turn this into a Nazgûl discussion, but surely they did not go around naked most of the time (even though invisible)! I'd always thought them to be clad in whatever they had been wearing when they had 'faded' when invisible to eyes (or as you said, like Frodo when he puts on the Ring). They were then obligated to put on additional garments in order to be seen by all but Sauron. Frodo, when wearing the Ring, was able to see their original clothes.

(Frodo) was able to see beneath their black wrappings. .......Under their mantles were long grey robes; upon their grey hairs were helms of silver; in their haggard hands were swords of steel.

FOTR p221. (paperback)
None of that was visible to any of the others present on Weathertop. Just the black stuff.

I fully agree with Morthoron about the wights. Most likely they were Houseless Elves inhabiting material corpses / bones - not their own, but those of the long-dead Dunedain.

Elves? It doesn't seem they were allowed to be ghosts hanging about in the ether. The spirits of all Elves went to the Halls of Mandos after the destruction of their physical bodies. And something bothers me about that arm. If it was an appendage of a long dead Dúnadan, why was it not skeletal? No- it must have been part of the wight. Look at what happened when Frodo attacked it.

With what strength he had he hewed at the crawling arm near the wrist, and the hand broke off; but at the same time the sword splintered up to the hilt. There was a shriek and the light vanished. In the dark there was a snarling noise.

FOTR p161 (paperback)

The sword broke just as it would have when striking a Nazgûl (though without the wounding of the sword-arm). The 'shriek' seems to indicate pain, the 'snarling noise', animalistic anger. Would any of that had happened if the arm had been simply a piece of corpse animated by another force?

And why didn't the Wights treat Frodo the same way they treated the rest of the hobbits? Was it because he was caught the last, or was it because the Wights felt he had the Ring and were going to deliver him to the Witch-King?

My only thought on that is that the ring must have been a factor, though I don't know that the wights actually would have even been aware of its existence. They were, after all, sent there by the Witch-king around 1670 of the Third Age- long after the Ring was taken from Sauron and believed lost forever. Perhaps the Wight sensed the Ring and was confused and alarmed, not understanding what it was, yet still desiring to kill Frodo along with the rest. Maybe the Ring also gave Frodo some measure of ability to resist the Wight's spell.
It's really all conjecture, I think, because there just isn't a lot of textual evidence to go on to make a point, one way or another.