Some really good responses here; this thread is getting very interesting! Keep them coming guys.

As I said, do elaborate further on your thoughts. It's all well and good giving examples, and that's great, but a really, deep, intriguing debate will come when we find something really powerful to discuss. Let's take this thread to the next level!
Hmm, interesting thread.
To answer the OP, I think the difference between Trek horror and your normal derivative "saw" style horror, is that Trek horror, for the most part works great at being psychologically terrifying.
You standard slasher film usually is more worried about how much it can gross you out.
There is a huge difference IMO. The morgue scene in Night Terrors is terrifying because the scene just gives you enough thread to tangle your brain in a "WTF is happening" thought. You see the bodies sitting upright, but still covered under the sheets. What's underneath? are the cadavers wide eyed open? Staring at Beverly? Are they all going to get up at the same time?
That's some scary shit.
Most horror movies now are just about how can we top the previous scene of a very brutal and gory torture scene, like how much more detail can we get to see of some guy's intestines being ripped open while he's alive or some shit.
That is not psychological at all. It's meant to trigger intense disgust and gross you the fuck out.

Well said! That is exactly the kind of thing I was getting at: psychological horror will always be scarier on a deeper level than the stuff which is simply gory. That may not be a fact per se - it's simply an opinion. But what if it is? Don't all the best horror films, including those with a really disturbing gory element, also have a strong psychological undertone or overtone? Is it not, perhaps, actually a fact that all horror, even the kind which is less psychologically slanted, still has a strong psychological bent? What would horror be without the psychology behind it?
These are the kinds of questions which I'm asking primarily because, as I said, it's fascinating how the more low-key and less cheesy or over the top something scary is - especially in TV Programs such as Star Trek - the more genuinely frightening and disturbing it seems to become. Again, should "Identity Crisis", an episode mentioned by several here, be scary? Well, thematically it's not far off from a cross between Alien, The Thing and some ultra-freaky human-becomes-animal (or in this case alien with the brain-power of an non-sentient beast) story. By itself, that should indeed be scary, and as an idea, it certainly is. But atmosphere is key in making this episode disturbing and uncomfortable to watch. It's played seriously, it's not over the top enough to be silly, and it builds slowly, so there is genuine tension, eerie tones and creepiness as Geordi slowly transforms into one of those creatures.
Any thoughts on why science fiction seems to be one of the best mediums for surprisingly subtle (sometimes), often low-key stories with a strong undercurrent and/or element of horror? You'd think sci-fi would be better at the more ridiculous and cheesy monsters, and it does that well, too - but, when it plays things more subtly, it can become really powerful and affecting.
Also, one other thing about Night Terrors: the morgue scene is horrible, but it wouldn't be so horrible in isolation. It's truly disturbing because it's one of the moments where (as with the snakes in Riker's bed, though that was significantly less freaky by a long, long way IMO) the viewer realises what the real horror is here: that the crew of the Enterprise are not only trapped by a strange rift with a ghost ship full of corpses for company - and whose people went mad and killed each other and themselves - but that they, too, will suffer the same fate. Dr. Crusher seeing the corpses sitting up as one, all covered in their death-shrouds, is awful, awful, awful. But that they don't do anything is even worse. Why? Because it's at this moment that we realise just how far the Enterprise crew have already gone. If, in such a short space of time, your crew is already hearing weird things, seeing ghosts, imagining snakes, and imagining the dead...well, moving...then how much worse could it get?
We don't see any hallucinations of a much worse nature than this, not after that moment at least. The point, however, is that because they can't sleep, the crew will lose their minds...imagine terrifying and bizarre things...and eventually go completely nuts and become psychotic. Worse of all, they're in space, adrift...so there's no escape. Imagine if the Enterprise crew had failed to escape, and then indeed suffered the same fate as the Brattain's crew, but one of them had survived...let's say Troi. Trapped alone, in deep space, aboard a ship full of your friends' and colleagues' corpses, with nowhere to go, no chance to escape...and your own mind will eventually drive you insane, too.
Permanent or near-permanent isolation is a scary thought as well. How many episodes across all the different Star Trek series, from TOS all the way to Enterprise, played with this concept? I'm pretty sure it was a hell of a lot...and yet they all worked, and felt neither derivative nor tacked-on, IMO.
Edit: One last addition: the scene where Worf is about to kill himself earns extra points for being, in and out of itself, one of the best examples of a scene in which nothing scary actually happens, but is still incredibly creepy and disturbing...precisely because nothing happens, ironically. Think about it: If Worf, of all people, the big Klingon guy (who himself is pretty scary in some ways) can be nearly driven to kill himself because he is terrified of something, but a something which is actually nothing, then you have to ask the question: what's scarier, a really dangerous thing which actually exists, or the idea that one of the toughest people aboard Starfleet's flagship can be scared of...nothing? The realisation here is a really uncomfortable one: here, all the monsters are within ourselves. The danger is losing our own minds because of what we fear might be out there...even if what's actually out there doesn't mean any harm. We don't know...it might do. It might be terrifying. And if it isn't, what do we have to fear most? Our own minds. Yeah, I'm pleased I wasn't aboard the Enterprise during that mission!