As Quizro points out, dark fantasy became regarded as being a sort of euphemism for horror in the 1990s. Why? Major publishers who flooded the market with awful, horrible, no-good novels in the 1980s to cash in on horror's popularity decided it was the genre's fault when readers were unwilling to buy mass-produced dreck. But, the market analysts noted, fantasy sold just fine. So the order came from down on high: market anything with otherworldly elements as a fantasy, anything with gritty crime elements as a thriller, and reject most anything else as unpublishable and let the small press handle it.
Both genres explore the nature of evil and the darker sides of human nature and create a creepy or frightening atmosphere. Thus, when asked what the difference between dark fantasy and supernatural horror is, some people will say that there is no difference, or that the difference is that horror goes to greater extremes of sex, violence, and, well, horror, than dark fantasy.
To my mind, that's a bit of an oversimplification: there are some books and movies that are squarely fantasies that are also really extremely gruesome and thus get sold as horror. While a broad gray area certainly exists between the two genres, there are a few ways to distinguish the two. (The standard disclaimer applies: these are general characteristics I've noticed rather than "rules", and those trying to separate the fantasy from the horror should look at a work as a whole, rather than latching onto a single element and thinking something like "Aha! Everybody dies at the end, so this must be horror!")
Horror is about an intrusion of the frightening and unknown into a mundane, everyday world the reader is familiar with. It doesn't have to be a present-day world, though; you can easily set a horror novel in a historically-accurate past. The intrusion doesn't have to be supernatural (a deranged serial killer will do just fine) though it often is.
Dark fantasies have an established setting that is fantastic or otherworldly. Such a fantastic setting can range from the overt sword and sorcery of Michael Moorcock's Elric saga to the subtle magic of many of Ray Bradbury's tales to the action-comedy of Buffy the Vampire Slayer. If you start out in a world where vampires or ghosts or magic are treated as a "normal" occurance by the characters, it's a fantasy world.
Book and movie series that start out as horror may then travel into dark fantasy genre because what was unknown and frightening in the first book -- say, a world overrun with zombies -- is established and known, though maybe only marginally less frightening.
The protagonists of dark fantasies are often heroic. They choose to face the dangers presented to them in the book, story or movie in order to save others or to achieve some greater goal. They are often experienced with the occult or in possession of special skills, knowledge, or powers. Clive Barker's private investigator Harry D'Amour (portrayed by Scott Bakula in Lord of Illusions) is an example of such a heroic character operating in a horrific dark fantasy universe.
The protagonists of horror stories and movies are often survivors. They're regular everyday people who have been thrust unwillingly into a frightening, awful situation, and they may be hugely unprepared to deal with it. But deal with it they must, or they die in often spectacularly nasty ways. Kirsty, the young heroine of Barker's novella The Hellbound Heart is an example of horror's archetypical survivor character.
In many dark fantasies, there's an implied comfort to the reader: the rollercoaster will stay on its tracks. The characters the reader cares about will usually make it out alive in the end, and the day will be saved.
Readers don't get that comfort in many horror novel and movies; the cars might go off the tracks at any time. The protagonist may surive the zombie hordes only to be shot by a redneck deputy in the final scene. Everybody might die. It's horror.
morven says: I do think that part of the use of 'dark fantasy' as a horror euphemism was a reaction to drummed-up outrage by Christian/conservative groups.
There is an element of that in there, certainly; horror has a reputation for being "nasty" and has, in the past, been accused of promoting Satanism because it explores the occult. I've met writers with a prudish streak who steer clear of horror simply because they feel it would somehow give them a bad reputation.
There's a long-established assumption in some quarters that science fiction (and, by extension, traditional fantasy) is "juvenile" literature, and thus is mainly reading material for teenagers. So, lots of magazines and other publishers have been historically reluctant to run works with profanity or graphic descriptions of violence or much sexual content. Much horror is squarely adults-only stuff that doesn't flinch from any subject or description.
However, in regard to publishers caving to protest from conservative groups, or that horror is especially vulnerable to it, bear in mind what's happened to the Harry Potter books. Nutters have been vocally protesting that Harry Potter promotes witchcraft, which therefore promotes Satanism. The protests wouldn't have even come up if not for the books' huge popularity, because magic is a staple in practically any fantasy novel I can think of, even Christian-influenced works like Lord of the Rings and The Chronicles of Narnia
Thus, I submit that most horror/dark fantasy stays off the nutters' radar because it's not the kind of thing they'd ever seek out to read, and the press doesn't draw their attention to it; they never were part of the market for these books, and publishers listen to the market.
Christian/conservative protests, in fact, have been good for book sales in some cases, because people run out to buy a copy to see what all the fuss is about.
It's when groups can exert sufficient pressure on local stores and libraries to keep certain books off the shelves entirely that the trouble starts (side note: don't buy your horror/dark fantasy novels at Wal-Mart: they censor them as they do their music CDs.).
But at least in the modern world, people can bypass local efforts at thought control and get their books online.