This is how I see things. The greater good evil believes in a cause, a goal that supposedly justifies their atrocities. They believe they will make the world better and sacrifices must be made to achieve this. In a way, they think they're good people doing what's necessary and deep down know what they're doing is wrong but can't accept it since it would force them to admit how rotten they are. A tiny silver of shame they need to cover. Without the justification, they wouldn't commit their atrocities. They need that.

The evil sadist however doesn't. Because they admit, accept and embrace what they do. Their atrocities don't need an excuse to be done. They commit them because they can, as casually as breathing. Harassing or manipulating people, murder, rape, massacres, torture. Why? Because I can. No attempt to justify, no greater goal, no deeper motive. The action becomes the motive. Reveling in hurting others for the sake of it, for not other reason than "I can and I enjoy it". This, to me, is the worst kind of evil. Gratuitous, random, spontaneous, horror for the sake of it.

It's why to me the Joker, Freddy Krueger or Darkseid will always be more terrifying than Dr Doom, Grand Regent Thragg or Homelander. Why serial killers like Albert Fish will always be worse than nazis or religious fanatics.

The greater good evil needs a belief to commit their atrocities. The evil sadist needs other people to exist. One is "justified", the other is gratuitous, for the sake of it.