Okay, so here's what I understand about the wights/White Walker issue...
A White Walker is a member of the ancient race (possibly dating back to the First Men? I'm not quite sure how old they're meant to be, but pretty old by this point) who are essentially immortal, except for using dragonglass or Valyrian steel against them.
(The show seems to have retconned this by adding fire as a weakness in Series 7, but I'm more partial to the books and 'retconning' itself to me signifies that either the writers have forgotten their original rules, or were uncomfortable with them at a later point, which is never a great sign, so I'm going to ignore it.)
The wights are essentially Westerosi zombies, as in corpses of dead people that the White Walkers have resurrected somehow. Depending on how long they've been dead, some appear more 'zombified' than others, yet the key feature is their glowing blue eyes, which at least is a constant in both ASOIAF and GOT.
(However, some of the wights in GOT seem to just have empty eye sockets, like this person here, so that's either a sign of decay in that their eyes no longer exist or... more retconning? Sigh.)
(Also, seeing as wights are sort-of zombies, does that mean they'll presumably die off by themselves if they decay for long enough? Seeing as some are already fairly advanced in terms of, y'know, being a corpse, would they just... fall apart? It's unlikely, seeing as magic could probably be used as a caveat, but it still made me laugh at the thought of a zombie shuffling along with its limbs falling off. Dark, I know.)
Basically, the difference between the wights and White Walkers (in my understanding) is that one is a pre-existing race and the other is magically resurrected dead people or animals - as in, they were initially normal and are now 'changed', whereas there is presumably no changing a White Walker into anything else. I'm not sure if the show will change this at some point, but at least that's how I understood it from the books.
And if D&D try to change anything else with their poor writing, I'll just react like this: