Why is the ending of Death Note in the anime so different from the manga? Even though I haven't read the end of the manga very carefully, I did notice that a lot of the details were changed, or even omitted completely, in the end of the anime. 

I don't understand why the end was changed in the anime. What is the reason behind the rather drastic differences?
As a few people said in the comments, it appears to be artistic license.

I'll mention first that, like @Deidara-senpai pointed out, there are some general reasons why an anime can differ from the manga. However, in the case of Death Note, there appear to be some unique specifics to their changes as well.

The anime director of Death Note partook in an interview in November 2007 for an issue of Newtype USA. (I don't have a link to an online version of this, I'm afraid.) He speaks a bit on why he made the creative decisions he did. From Wikipedia:


  Tetsuro Araki, the director, said that he wished to convey aspects that "made the series interesting" instead of simply "focusing on morals or the concept of justice." Toshiki Inoue, the series organizer, agreed with Araki and added that, in anime adaptations, there is a lot of importance in highlighting the aspects that are "interesting in the original."


They go on to speak about a few specifics, such as Light's presence in the anime versus in the manga, but in general they also speak about the logistics of converting the manga into an anime, which is a challenge itself:


  Inoue noted that, to best incorporate the manga's plot into the anime, he "tweak[ed] the chronology a bit" and incorporated flashbacks that appear after the openings of the episodes; he said this revealed the desired tensions. Araki said that, because in an anime the viewer cannot "turn back pages" in the manner that a manga reader can, the anime staff ensured that the show clarified details. Inoue added that the staff did not want to get involved with every single detail, so the staff selected elements to emphasize. Due to the complexity of the original manga, he described the process as "definitely delicate and a great challenge." Inoue admitted that he placed more instructions and notes in the script than usual. Araki added that because of the importance of otherwise trivial details, the notes became crucial to the development of the series.


As you can see, the director and organizer of the series seemed to agree that a slightly different feel was needed in the anime, so artistic license was taken. Unfortunately, they don't go into huge detail over the specific changes (except for Light's own characteristics), and particularly not over the ending. I can only surmise that it was altered to achieve the feel they were generally looking for in the anime.