Mostly art things. I'm far from qualified to speak to it as an expert, I haven't played either version yet, but have friends who are very passionate about the topic.

I think the easiest way to explain it is to refer to this Miyazaki quote:

Most people don't believe me when I say this, but a certain kind of refinement, elegance, and dignity are very important to me. I'll usually tell the designers that flat-out grotesque or splatter type designs will not get past me. This has everything to do with my own personal sensibilities, and it is something that I apply to every design that I approve.

Even if you just look at the tutorial boss (I clipped a a YT side-by-side for you here), the changes they've made here to add detail are basically all... grotesque. Gross hanging flesh, some weird hanging nipple thing, it's a very different interpretation of the original than what I believe was intended.

This is obviously just one example, but it's this type of change that bothers purists. Now, mind you, I don't think this makes the remake trash or anything, but if you're interested in Demon Souls historically as the beginning of the Souls franchise, this kind of change is essentially revisionist history, and it's disappointing to me that the original game isn't also available in some way besides buying an old PS3 or emulating the game.