Here's a few reasons:
1) Different visions. Often, the manga and the anime of the same franchise will focus on different aspects of the same idea. This can be by intent - Revolutionary Girl Utena, for instance, was created by Chiho Saito (a mangaka) and Ikuhara (an anime director) as an outline, and then each took that outline and produced their vision of it. Chiho Saito wrote a very tight manga about magical duels at an otherwise normal school. Ikuhara delved into reincarnation, purgatory as a boarding school, and an exploration of how we understand reality. He expanded several characters greatly (including one character who was based on a photo in another character's room). The end result was the same basic story told in vastly different ways, with different characters (who happened to have the same names - Manga Juri is nothing at all like Anime Juri).
Two other less extreme (or planned) examples are Mai Hime and Evangelion.
Mai Hime was a shounen harem manga. However, the anime it was made into was a shoujo with some shonen overtones (mostly some breast-based fan-service that gets (mostly) forgotten after the first four episodes or so). The main character of the manga is the male lead - his part is vastly reduced in the anime, so that he isn't even considered one of the three most major characters in the anime. For this (and other) reason(s), the end of the manga and anime are very different.
Evangelion, on the other hand, was an anime first and a manga second. The manga, written by a different author who worked with the original anime author, chose to make the manga a slower, more character focused piece. He added a lot more character development to the series - and the original creator loved his work so much that he incorporated several parts of it into his remake ten years later.
2) Plot variation. Less drastic than the above, but related. Often, during adaptation, changes will be made. Others have noted this above. Typically, if I like a manga, I seek out it's anime - and vice versa. I then compare the two. In several instances, I've found one version of the story superior to the other. Examples:
Kashi Mashi. The original manga has a wonderful story that is well paced and makes sense. The anime scraps the original plot and instead chooses to be both cowardly to the subject matter and illogical at the same time. It was so awful, I sold the DVDs (which I bought on faith because the manga was so good).
A less horrible example is Love Hina. The manga is all over the place and doesn't really know where it wants to go. The anime, meanwhile, while guilty of some very over-the-top padding, actually paces the story well and ends in just the right place. The movies that follow kinda ruin it (and the less said about Again, the better), but just taken on the anime itself, it has a very tight narrative that doesn't outstay it's welcome.
3) The manga and anime are different parts of one story. Two examples will explain this best: Ghost in the Shell and Gunsmith Cats.
The Gunsmith Cats manga is a specific story that has a beginning, middle, and end. The anime, meanwhile, was specifically designed to fit "in between" manga chapters and not have any affect on the plot. Basically, the anime is nothing more than filler episodes to the manga.
The story of the Ghost in the Shell manga is very complex. The famous movie is actually two chapters out of the middle of the manga, and one chapter from near the end all slapped together. If you ever watched the movie and wondered why the hell they were doing what they were doing, that's why. If you read the manga, you get the introduction of the characters, the slow lead-in to the main antagonist, and then the plot cranks up to where it is in the movie. The manga is paced (and handled) much more like Stand Alone Complex (including the humor).
Stand Alone Complex is an excellent anime, but again it is designed to fit between chapters of the existing manga without making any waves. The beginning, middle, and end of the story have already been written - SAC, while awesome, is essentially a filler arc to the manga.