Most intellectually-minded people will insist that the book is usually better. I usually like to watch the flick first and then read the book, because it allows me to view the two as separate entities. On the whole, I get personally attached to books much more easily than films (although I am a film lover, as well). If I see the film first, I can enjoy it without missing my favorite parts from the book that they didn't include. It also makes reading the book more enjoyable, because it's always exciting to find the extra little story facets that books invariably include.
With short stories that are adapted for films, it gets a little dicey. Often, the short story can easily stand on its own, and the extraneous story added to make the film at least 90 minutes seems like filler. It's very hard to turn a short story into a movie and do it right. Two of my favorites are both based on Stephen King stories, The Mist and 1408.
King's story "The Mist" is a solid, creepy story that uses a menacing mist and sci-fi creatures as a smoke screen for a good old-fashioned claustrophobic cabin fever mob mentality tale.
Along the same line, Frank Darabont's (of The Green Mile and The Shawshank Redemption fame) The Mist looks like a high-budget sci-fi creature feature, but is just as compelling and well-acted (especially by veteran actress Marcia Gay Harden) as any of his other films. It jives with the original story and doesn't make you feel like you've watched 30 minutes of decent plot packed with 60 minutes of fluff. It is everything you expect from an excellent adaptation.
Even more rarely, you get an adaptation that surpasses the story, which is the case of Mikael Hafstrom's 1408. King's story is essentially about a jaded writer who writes about haunted places, but doesn't actually believe in ghosts or spirituality. He visits a supposedly haunted hotel and believes he'll have a routine stay, but is overcome by the unexplainable happenings in his room. For King, it's a fairly subtle story, with few of his normal shockers. It's a good story, but not one of his best.
The film, on the other hand, seems more like King's work than the original. It's got the black humor and dread-inducing creepy moments that you've come to expect from the Master of Horror. Of course, the film's quality is mostly due to its two principles, John Cusack and Samuel Jackson. Cusack is pitch-perfect as the talented writer turned tortured skeptical hack, but it is Jackson's performance that makes the film refreshing. He plays an urbane and vaguely menacing hotel manager who warns Cusack's character several times to avoid room 1408. Hafstrom is good in his own right. He plays some interesting games with the audience, including making us wonder if Jackson's character is truly the manager, or a product of the room (reminiscent of The Shining). The use of The Carpenters' "We've Only Just Begun" at pivotal scary moments is also a killer move (I'll never think of that song the same way again). And, while the original is good, the film is much better. (I would reccommend Hafstrom's original director's cut ending rather than the theatrically released finale.)
Check out both stories and both films. Even if you disagree with me one way or another, you won't be disappointed.