I like to play it like a X-files movie with less supernaturals. It can be played like any modern film, that's one of the strengths of the system. There is magic and dreamlore which allows for a Terror on Elm-street feeling if you're in that mood, and so on.
I rather try to play it like a succession of shock sequences, more and more strong and horrible, starting to use the supernatural only when the shocking power of "natural" things is exhausted in a scenario... This gives an extra boost to the horror, at a point where the players think you're at the climax, and I like to show them that we can go much farther...)
I play with the unstable reality, the time and the space often mess (near the climax of the scenario), and I play on the most paranoid of my players...
Unfortunately, I play only "stand alone" scenarios because I don't get the players for a campaign game (most of the players around me don't even want to hear of Kult). Maybe I'm using this kind of "shock scenarios" where players are mere survivors because of these "one night" basis games... I lack the experience of a Kult campaign, which would be interesting to follow the transformations of the characters and their progression on their path to Awakening...
The biggest difference from other games is that it's present day horror where you're the weak part. I think lots of horror games have been written with that intention, but players being what they are, the nature of the heroic idiom etc. tends to lead to games where players become larger than fiction. We've all seen it I'm sure...
Player- Whaddya mean you're the all powerful Cthulhu and you want our obedience? Give the virgins back, then Piss off, creep and take those hideously writhing tentacles with you!
Cthulhu (looking browbeaten and sheepish)-er, Oh sorry, er, ok then, sorry. (exit cavern left)
Player- Jeez, what a loser! Now where did that scum-sucking Nyarlathotep get to?...
Most of the time it's the humans that's the horrible monster and your struggle is both physical and mental since you have to fight your fears and darker sides as well as baddies.
Vampire has a couple of the same elements, but not to the same extent. A vampire is only scared of sunlight and his prince, more or less. A Kult person is scared of the same things as you, which could be anything.
I think that depends on how the ref and players gel, and what they want in the first place. Some folks like being monsters (sure, I do once in a while) while others really only like to fight against evil / the bad guys. I guess this might make Kult or any "monster within" game tricky for some peoples sensibilities.
An excellent place to get a feel for the true unpleasantness of the Kult world is the published scenario "Taroticum". With only moderate tweaking, this became one of our most talked about adventures: the general feelings of the players seemed to be "Wow1 That was great, what the fuck is going on?!"
I've found this, for our group, to be the essence of the game, a feeling of total helplessness- thrust into an accelerating spiral wherein only the tiniest threads of reason connect. This exacerbates the characters phobias and Dark Secrets, especially when played against their mundane lives within the illusion.
Gradually, as they gain experience, they may see some of the true reality and, should their sanity withstand the assault, they start to understand the total futility of existence, the first step to Awakening (but the hardest to overcome). A slowly unfolding psychodrama, with only the very occasional supernatural entity to start, with more introduced as the months go by.