First thing to remember: the short story is not the screenplay.
The short story is not bad, but it's not the same.
King's story was creepy in its own way, and it deserved to be made into a movie. I actually can envision a story written just on the entries of Mike's minirecorder. However, I would have preferred that to the skipping between what was on the tape and what the lead character experienced — at least the way King wrote it.
I also preferred the way the movie ended to the story's ending. (Tip number two: don't watch the alternate endings on the DVD. There's a reason they weren't chosen as the final ending.)
The movie, however, included more about the lead character, a writer whose own tortured soul was mixed with the unrelenting evil of the hotel room. The movie wasn't perfect: it could have been a little shorter, and the hotel room scene went beyond the point of exhausting to making me wish it was just over, for the love of Pete (and not in a good way). Having said that, I wouldn't really opt to change much.
I am thrilled that the producers chose the actors they did for the movie. King's descriptions do not jive with the actors, but I preferred the actors to those described in the story.
If I had to choose only one, watch the movie and skip the story. You lose nothing by seeing only the movie, which captured the good parts of the story. And if you are going to do both, watch the movie first.