Question 1: "what do the mechanics do that is interesting and reinforces the theme of personal horror?"
Question 2: "what do the mechanics do that is interesting and storytelling play style?"
I can't speak to the older edition of World of Darkness games, but my overall impression of the newer ones is that the focus on personal horror, or whatever the particular sub-theme of splat may be, has been greatly tightened up and distanced from the trend of 'superheroes with fangs and trenchcoats' that the old World of Darkness games had a way of falling into.
One example of a newer WoD line with a very tight focus is Promethean: the Created. The general premise is that one plays a revived corpse of some sort, an abomination of nature. Essentially, you're running around as Frankenstein's monster in disguise. The proposed 'goal' of any given character is to complete a successful Pilgrimage, with a pretentious WoD Capital Letter. This is essentially a journey and process of self-enlightenment and refinement, with the end goal to be come a mortal natural being again with memories of the person you once were. Promethean is heavily influenced by old ideas about alchemical philosophy and self-refinement, and the different paths one can take on one's Pilgrimage and the associated nifty powers all reflect this.
But as a hideous abomination of nature created from a dead man's corpse, you cannot be allowed to hide away and study in peace. As Frankenstein's monster, you need to be chased out of town with riots and lynch mobs and flames. This is mechanically represented as Disquiet, the ranked from 1-10 stat that measures how pervasive the unnatural aura around you is. At low levels people may simply react with distrust and animals will run away from you or bite you. High levels of disquiet result in sudden irrational hatred and violence. Yep, there's your lynch mob. Even a reclusive Promethean that only associates with other mockeries of life is encouraged to stay on the move, wandering from place to place, because staying in any one place for too long creates a supernatural Wasteland of urban slumland, dangerous wilderness filled with feral beasts, etc.
So the overall result is a lonely existence, wandering from place to place out of either a sense of responsibility or a lack of welcome. It should also be noted that the rules for how a Promethean may complete a successful pilgrimage exist in the main splatbook and are laid out in detail, as opposed to vague ideas about how a vampire might reverse their own condition.
There are a lot of other ideas I want to comment on while I've watched this debate, but I've got to get to my doctor's appointment. I just wanted to provide a brief refutation of the idea that WoD games don't provide any mechanics that aid in genre emulation.