One big difference is their origin, what they were and why they were in Middle-Earth. The Nazgul and Dead Men of Dunharrow were originally living Men. The Barrow-Wights were some sort of "spirits from Rhudaur and Angmar" (The Appendices to LotR) who entered the empty mounds (though no one says they could not have been spirits of evil Men from there - recently there was some debate about this and someone mentioned that in HoME there was a variant of Sauron trapping spirits of some, but I believe these were Elves, and forcing them to do his bidding - therefore the "Necromancer" nickname). Also, the Dead Men were undead because they did not fulfil their oath; as a result of curse. The Nazgul were undead because of their "fading" after wearing the Nine Rings. The Barrow-Wights... well, who knows. If they were originally just "spirits", then they might have inhabited the wraith-world from the very beginning and therefore, calling them "undead" won't be as "politically correct". Though there is the possibility that a Barrow-Wight, in fact = some empty shell of a skeleton + some sort of a said spirit to give it a "life". So Barrow-Wight as a whole could be called "undead", while the original spirit may be not. But that's just guessing.
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"Should the story say 'he ate bread,' the dramatic producer can only show 'a piece of bread' according to his taste or fancy, but the hearer of the story will think of bread in general and picture it in some form of his own." -On Fairy-Stories
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