tl;dr : It's a mixed bag. Overall, the bad outweighs the good, but there are some good changes in there. And some very, very, inexplicably bad ones.
Strange looks godawful. The upgrade managed to make his beard and eyebrows look shittier. And what's up with the reflection in his glasses? In the original you could see Bruce's face, in the remaster you can't.
Penguin and Ra's look awful. They tried to brighten the colours on Ra's tattoos in the remaster and just made it look cartoonish. His face also looks much less sinister.
Scarecrow looks terrible. His mask looks like it's made of yellow rubber. Not to mention the overall brighter lighting works to the detriment of the creepy scarecrow man.
Joker looks mostly the same except for the much brighter colours on his clothes and hair. I'm not sure if I like it, but it's a stylistic choice and not a matter of quality. His eyes are a much brighter green, however, which I love.
Two-Face looks basically the same but, in an inversion of the general trend, is much less well-lit.
Poison Ivy looks...not much different, but still kinda worse. The plants have a cheaper look to them and her hair is much less vibrant for some reason. Her jacket does look better in the remaster.
Harley looks better. It's probably just the brighter lipstick and the slight smile, but I like the reshape of her eyes and the brighter colours in general. The only thing that looks worse in the remaster is her collar, which looks badly rendered.
Batman looks way worse...they did something to his face? He looks like someone just shat in his Batmobile. The rain and snow effects on his cape are more advanced, but somehow make him look worse. The detail on the costume is slightly better, particular on the wrists and shoulders. When the cowl is off, his hair looks much, much worse. The original actually looked like hair.
Catwoman looks better. The darker costume and makeup has a good contrast with the brighter lipstick.
Freeze looks a little better. Better lighting works well with his suit.
Robin looks...a little better?
The shot of Bane has this bloom effect that obscures most of the detail. Not sure if that's indicative of the overall quality. Killer Croc has exactly the same problem, but with water effects. It just makes it harder to see what's there.
Clayface looks orange as opposed to brown. I think that's an effect of the overall brighter light.
Overall, the big changes they made were adding some fine detail to the costumes, adding weather, water and glare effects, and brighting the colours across the board. It sometimes works and mostly doesn't. The brighter colours and lighting sometimes serve to make the models look cartoonish as opposed to sinister or menacing, and when the glare isn't making it harder to see, what you're actually seeing doesn't look good. I think this is a stylistic complaint, but overall I preferred the claustrophobic, poorly-lit nature of the originals.