The degree to which players become more powerful as they level in D&D's 5th Edition is pretty huge. As casters get access to more powerful spells and martial characters get various ways to do more damage with their attacks - or get broader access to magical weapons - player characters start to carve through foes that would, earlier in their careers, have been major problems.
I actually think that the clearest way that characters become more powerful is simply their rising maximum HP. The Commoner stat block is meant to represent a typical ordinary civilian of average ability, and they have 4 hit points. While this looks really bad, consider this: a dagger is one of the least powerful weapons in the game, dealing only 1d4 damage on a hit, plus modifiers. If you were to walk up to an ordinary person and stab them (which, you know, please don't. That's attempted murder at least,) that would be a very significant injury. If you hit them in a vital organ, like the heart, lungs, throat, or a major artery, you could kill them - which is basically what would be the equivalent of rolling a 4 on the die.
But even the squishiest of D&D classes, the Sorcerer and Wizard, start off with, at minimum, 6 hit points (and likely more, as either of these classes should probably invest a bit in Con for both HP and concentration,) meaning that unless they crit, a commoner can't take them out with a single stab with a dagger. And that's just at level 1.
If you take the "standard" HP boost with each level, you get to add 4+Con every level, meaning you've got at minimum 10 HP by level 2 (assuming you don't have a negative Con modifier, and I cannot imagine anyone taking a negative Con modifier on a character they want to, you know, survive.) If you roll your health, you're going to have at minimum 7 HP, or on average 9.5, and if you do the Mercer-style "roll but you can always re-roll 1s" style, it's an average of 4 on the die.
(As a side note, I just realized that Mercer's style does actually give you the same average as the "standard" amount. I think I had previously treated rolling 1 as a single re-roll that could still give you a 1, rather than basically ignoring 1s and re-rolling until you roll something else. Another way to think of it is this method turns 1d8 into (1d7)+1)
Anyway, the point is, let's imagine a classic slasher killer like Jason Voorhees. This guy's definitely stronger than average, maybe even really at the pinnacle of human strength at +5. And his trademark machete is, we'll say, the equivalent of a longsword. That means that when he hits an unsuspecting teenager (who would likely have the commoner stat block, though maybe with a -1 or -2 to Intelligence and/or Wisdom) he's doing, on average, 9.5 damage - more than enough to kill his victim outright - no death saves, no nothing, just a splatter of blood and a bunch of trauma for the little kid who wandered into the TV room while his teenage cousins were watching a movie (genuinely not an experience I had, but it feels very right for my age group, who always wanted to seem cool to our older, Gen-X cousins.)
But as we saw, a 2nd level Wizard is already at a point where they will usually survive a slash from Jason, even if they have a +0 to Con. And that Wizard also probably has a Fighter or Paladin friend who can then attack back at Jason, giving the Wizard time to hook their thumbs together and burn the crap out of Mr. Voorhees with a bunch of fire from their fingers.
By level 5, our, again, shrimpy, +0 Con Wizard has 30 health, meaning that Jason's got to get them on average four times to take him down, and I don't think Jason even has multiattack!
And therein lies the problem:
Stephen King defines three types of fear: gross-out, horror, and terror. I'll just stand on my soap box and say that I think his definition of horror and his one for terror should be swapped, but I'll use his definitions just because, well, Stephen King is a much more prominent authority on the subject than I am and more people are likely familiar with his terminology than what I think it should be.
Gross-Out is just gory nastiness - bodies being ripped apart, oozing blood, rotting corpses, etc.
Horror (in his terms) is a clear and obvious threat coming at you - a monster charging you or grabbing you with its claws.
Terror (again in his terms,) though, is a creeping sense of dread, where ambiguity leaves you frozen in the state before your animal brain chooses between fight or flight, where you might not even be sure if there is a threat yet, but you feel afraid. This is the creepy house where you could swear you hear whispering coming from another room that's supposed to be abandoned, or a dark basement in which a strange man is standing facing the wall and won't turn around to face you even when you say hello.
Stephen King's terror is really what makes the horror genre tick - the anticipation is the mood that a good horror story focuses on, those ambiguities in which you're hesitant to turn the page (or continue watching the scene in the case of movies). Paradoxically, once the horror starts - the monster jumps out - the tension is actually relaxed. Yes, you might flinch at the very moment of the jump-scare, but this is generally when the music (and soundtrack in general) gets louder, and the focus is action rather than mood, where either the character is desperately running away from the monster or is now being brutally slain (you can get some suspense out of these sequences by making it an open question if the character will make it out, but it has a very different feel.)
(Ok, one more little soapbox moment: the reason I think "terror" better describes what King defines as "horror" is in part tied to my having spent much of my teen and early adult years in the War on Terror era. When I think of "terror," I think of people in downtown New York running for their lives as the WTC towers collapsed - a very clear and obvious danger. By contrast, when I think of horror, I think of the horror genre, and as I previously described, I think that said genre works best in its eerie anticipation.)
In a D&D game, however, regardless of what class you're playing, the type of character you play is one who has basically made a career out of fighting monsters. And, to be frank, I've always felt that the fantasy and horror genres are actually pretty much the same, except for some differences in tone and expectation. Fantasy is horror in which the protagonists are well-equipped to fight the monsters. Your heroes in fantasy are not the in-over-their-heads teens at Camp Crystal Lake, but are instead trained knights, mages, and martial artists who know exactly what they need to do to kick monsters' asses.
Indeed, if you look at the Evil Dead trilogy (which is really weird because the second movie is more of a remake than a sequel) the first one is kind of your standard "teens go to a remote location and are picked off one-by-one by supernatural evil" kind of 80s horror movie, with a very strong emphasis on gross-out effects. In the second, the breakout character of Ash turns the tables by the end, becoming a monster-hunting badass with a chainsaw attached to the stump of his severed hand.
By the third movie (my favorite by far,) Army of Darkness, the series has transitioned entirely to fantasy, even transporting to the Middle Ages, where Ash leads the defense of a castle against an army of skeletons.
Basically, I think that a D&D horror game is, in almost all cases, going to be distinguished more in aesthetic than actual mechanics. The party is always going to want to, or at least try to, fight the monsters that attack them, rather than always run away. They're not commoners, whose smartest move is to get the hell away from that hockey-mask-wearing, machete-swinging brute.
You could say that a horror version of the game will be balanced against the party - far more deadly encounters, and enemies that the party will have a very tough time facing, which might force them to retreat.
But that's not really outside the realm of fantasy either. If you look at Lord of the Rings, the Fellowship's pursuit by the Balrog is absolutely such a situation - this guy outclasses us, so we gotta run. But that sequence, even if there's a sense of dread and doom, does not have the same feel as a horror movie.
Furthermore, the sequence at the end of Fellowship in which the Uruk-hai attack is similar - the Uruk-hai even manage to kill a party member. Indeed, the Uruk-hai are more or less an entire race of Jason Voorheeses - similarly violent, similarly tough to kill... for an average person. But by that point, the party is basically all at least level 4 or something, so they can carve through orcs with ease. It's really just through overwhelming numbers that the orcs are able to take down Boromir.
I think, thus, that the key to making a horror D&D game is to really emphasize King's "terror." The "jump scare" moment is going to turn into standard D&D combat one way or another, but the build-up to it will be what gives your players a fright.