I think your question is closely related to questions like Why aren't books adapted one-to-one?.
One aspect why an adaptation is different is because there are other people working on it.
If you want to express feelings, there are many different ways to do that. For example, the mangaka thinks that a "No..." with a sad face is enough emoting while the director of the anime thinks this is not enough and has the character do a Darth-Vader-like "NOOOOOOOO!!!!". One example of this is (not an anime, I normally just read manga) Harry Potter and the Goblet of Fire. In the book, it says:
"Did you put your name in the Goblet of Fire, Harry?", he asked calmly
while in the movie we get this.
On the other side, you have a book like "Who censored Roger Rabbit" by Gary K. Wolf, which was adapted into the 1988 Who framed Roger Rabbit. In the book,
Roger is killed in the beginning of the novel.
Following this event, the detective tries to solve this mystery instead of the one he tries to solve in the movie. Thus the plots are completely different and bear no resemblance to one another. The adaptation is, so to say, in name only.
This might happen because the director thinks that he has a better grip on the story the author wanted to tell, so he just "improves" it. Or he thinks that he has a better story to tell in the universe and he just wants to import all of the framework the title entails. This can be seen with video game movies: the Resident Evil movies import nothing from the games except maybe the premise of zombie-like creatures attacking and some names.
And then there are those directors who just want to profit from the name. Their story has pretty much nothing to do with the original material. For example, the Tomb Raider movies contain no levels from the games (that were released before the movies were made, don't know about newer ones). Lara could just be called Nancy Drew and could be a 13 year old girl; except for the sexualization, it could be the same movie. Lara could have been a 40 year old male "archeologist". But the studio wanted to profit from the name "Tomb Raider", so they made a Tomb Raider movie that had nothing to do with the Tomb Raider games.
PS: The last two categories can be combined. You could pretty easily swap the Resident Evil and Tomb Raider examples in them but the motive is different.