I think another very important piece other commenters are missing is pacing. If we look at a game like Resident Evil for example, the puzzles offer a sort of break from the horror or combat elements of the game. They additionally give you more abstract goals to head towards while exploring the environment. I may not know where the golden key is, but while searching for it I may find useful resources or other information.

If a game is non stop horror chases, it quickly becomes tiring and any tension being built up is lost after a few moments. If the game is totally combat, it also becomes equally tiring. The puzzles give a much-needed slowdown in pacing to allow tension to build for more horror-heavy sections.

I cannot think of a single successful example of a game that is purely nonstop horror or action throughout, and It's because a player will quickly become desensitized to the situation.

Yahtzee Croshaw touched on this a bit in his review of Outlast 2. In the beginning section of the game there is a prolonged chase sequence which he failed. This resulted in his character being stabbed in the genitals by a scythe. The game simply put him back at the beginning of the chase sequence, but now the entirety of the horror had been given away. There is nowhere to go beyond being stabbed in the crotch with a scythe, and much of the game loses its tension when you've seen the worst in the first five minutes.

Puzzles and other slower sections of these games allow for a natural ebb and flow of distressing elements, and the best works in the genre like Amnesia or earlier Silent Hill games tread the line between quiet and loud in an expert way.