It's like damage control for the mistakes of the second film. (Spoilers)
Hellraiser: Part II was virtually a retread of the first film, but it made some mistakes along the way. The part about the creepy uncle being tortured with sexual denial, Pinhead being revealed as formerly human, the defeat at the hands of the doctor cenobite, and the goofy ending with the distorted box were all mistakes in my opinion.
When I saw that goofy giant box with multiple faces on it, I thought to myself that the third film could only ignore the ending to the second one. Instead, they brought the box back and redesigned it to be a distorted art piece / prison for Pinhead. The backstory with Pinhead being human was once more brought up, but this time fleshed out. I also liked how the film turned the soldier and the demon into two separate entities, the latter being an offshoot of the former.
I'm surprised that this film has a lower rating than the second one. I guess it's because it's not as gory and the film takes a while to get going. Overall though, I found it much more satisfying than the second film.