Via Author, I found this question posed by Iwa ni Hana:
Why would fans want to experience / creators want to tell more or less the same story with more or less the same characters in different formats, be it manga, TVA, OVA, feature film, CD drama, novel, live action movie or live-action TV series?
The structure of the question pretty much demands a two part answer (one for fans and one for creators), and I'll tack on some tangents while I'm at it.

I imagine that the creators question has the easier answer, though there are really several possible reasons why a creator would want to adapt their work to other mediums. Perhaps the creator always wanted to make a movie, but lacked the resources and expertise to create one, so they started with a comic book/manga/web comic instead (Author notes this in his post - "formats form a vague hierarchy of expense, with cheaper works (such as manga) forming the base and being adopted into more expensive arts."). Another big reason could be because the creator wants their story to reach a wider audience. A corollary to that would be that the creator would assent to an adaptation because they were paid well, and if the adaptation is successful, they may be able to achieve a higher degree of independence or creative freedom in their future work. Note that these aren't necessarily good things, but high-cost mediums like film require creators to make a name for themselves before studios will sign off on the budget for a dream project.

This probably isn't that common a scenario, but it's definitely possible, and the history of film shows great filmmakers "slumming it" before they go on to make their classics. Take Stanley Kubrick. He got his start as a photographer for Look magazine. He once did a photo-essay on a boxer named Walter Cartier, which he later adapted into an independently financed short-subject documentary called Day of the Fight. He parlayed that minor success into a few more short documentaries and then into narrative fiction films, doing kinda standard noir thrillers like Killer's Kiss and The Killing. These are fine films, and better than most of their contemporaries, but Kubrick was also paying his dues in the film industry, which is something he continued to do up until Spartacus, after which his career really took off. He had proven himself a bankable commodity. A filmmaker popular with the critics and with audiences (a rarity, to be sure). Again, this probably isn't that true of all artists who do (or allow) adaptations of their own work, but it seems likely that at least some creators would pursue other mediums so that they can tell the stories they want to tell.

The fan's perspective is a little more complicated. Why would you want to watch what basically amounts to the same story you just read? I'm honestly not sure. Personally, there are definitely cases where a book is adapted into a movie and I dread watching the movie (said dread is often justified). But there are a few reasons this could happen. First, it could be a way to introduce a friend to one of your favorite authors or books without nagging them to read the books. Second, there is often a chance, however slim, that the adaptation will add something new and interesting to the source material. Most adaptations are, by necessity, not the exact same story. In the rare instances where they are, they generally turn out a little bland (I actually enjoyed the first two Harry Potter films, but they're also bland and a little boring if you've read the books). Indeed, many of the best adaptations are significantly different than their source material. Not to keep using Kubrick as an example, but The Shining is a wonderful example of a movie that only bears a superficial resemblance to the book, and yet is quite entertaining. It's also one of the few examples of an adaptation that has carved out it's own reputation without affecting the reputation of the source material. In my mind, both the book and movie are classics, but for different reasons. This actually makes sense, as different mediums use different "language" (for lack of a better term) for telling a story. I think this is part of why authors who write the screenplays for movie adaptations of their work often produce disappointing results. For example, take any number of Stephen King adaptations where he's written the script, including even The Shining mini-series, which pales in comparison to Kubrick's film.

This brings up an interesting question about movies that end up being better than their source material. Of course, most often, it's the other way around, but in some instances, lightning strikes. Unfortunately, I haven't read many of the typical examples, but from what I can see, both Jaws and The Godfather took rather conventional source material and elevated them into classics. One I have read that's a better movie is The Bourne Identity. It's not an utterly brilliant movie, but I thought the book was poorly written (though I think I like the story better). Other books I've read that have at least comparable or debatably good adapatations are Fight Club and The Exorcist.

All of which makes me wonder why people don't adapt (or remake) bad stories that have a neat idea. The All Movie Talk podcast had an interesting list of movies that should be remade, and I think it's an interesting concept.

But I digress. Another reason fans might want to see an adaptation is that they're just so enamored with the characters or the story that they revel in any chance to revisit them. As Author notes, other mediums may add something of value to the original work, even if the adaptation is not as good as the original.

So to recap, there are lots of reasons! Personally, I find the most compelling to be spreading the story around to a wider audience, though I do have a soft spot for wanting something new and exciting from an adaptation. Then, of course, you also get totally off the wall stuff like the movie Adaptation, which is based on an oddly recursive story: The screenwriter, Charlie Kaufman, was hired to write an adaptation of Susan Orlean's novel The Orchid Thief, but he found the task to be quite difficult and could not seem to make any progress. So instead of actually writing the adaptation, he writes a script about how he is having trouble writing the adaptation. (A quick tangent: Ironically, the one story that Stephen King has sworn not to sell the film rights for is the Dark Tower series, in which King basically pulls the Adaptation trick.) In the end, I think adaptations are good things, even if many of them are of dubious quality.