Reading this one was a life changing experience for me. I was in a bit of a dark place for a few days after reading it, since I wasn't really sure how to view the world anymore. The story is about Satou Tatsuhiro, a social recluse who is deathly afraid to leave his house and tries to reintegrate himself into normal society. The characters all have their own problems as well, in an Evangelion-like fashion. There is also an anime, though the story branches massively from the manga, and is a bit more light-hearted while still carrying many of the same themes as the manga.
Overall - the manga I personally give a 9.5/10 and the anime I give an 8/10. Both recommended.
As a note: this manga is VERY ADULT in nature, with sporadic nudity and major references to sex. It is also depressing as hell.
>>2 If I remember correctly I think it does parts out of order and sometimes differently in the anime. I do know the manga has extra parts the anime doesn't but I wouldn't really say that there's a point where it strays from the manga. It's been a while since I read it so I'm not entirely sure though.
>>3 That is correct. The main story in the anime is similar to the manga but does some scenes very differently/more lightheartedly. Around episode 20 or 21 the ending completely changes, and the manga is only about a third done at that point.