This game is about shooting demons in the face. And it's good about it.
That's the short story.
The long story, however, is that Painkiller: Hell&Damnation is a straightforward old-school FPS with a bit poor storyline, but it's fine, because you get a lot of action instead. I started to play this game for this particular reason, because I was tired of boring side missions in Borderlands 2, where you need to collect cookies or walk a dog. I just wanted to shoot things in fast pace and this game offers precisely that!
I very much enjoyed the demonic graphics and infernal landscapes, as well as the boss fights. Though, this game runs short of bosses - they're only four bosses in the main game. I particularly liked the Orphanage, which was really spooky - I'd say it's the second most spooky level in my experience after Return to Castle Wolfenstein's Catacombs level.
As to the co-op experience, it's decent, but it's lacking mini-map, so it's hard to locate your partner.
Also, I think that Painkiller: H&D has one of the best soundtracks in gaming history.
After you're done with the main story, you can buy DLCs, play community maps or try find the secret areas and holy objects, which is not easy. So there's a certain degree of replayability.
Some complain that this game is a downgraded remake in terms of the content, which I guess is true and there is a lot of (and expensive) extra content for Painkiller, but since I bought Painkiller for 4 EUR in Humble Bundle, I have no complaint. I'd recommend to buy this game with discount.
So, to sum:
lots of action and shooting;
decent amount of awesome weapons;
hidden areas and secrets;
no mini-map in the co-op mode;
it takes a bit more time than usual to understand how this game works;
My overall rating: 8/10.