Anime and manga are intertwined. The majority of anime are based on manga, and original anime are often turned into manga.
Japan has a different comic environment than the United States. Manga is ingrained in the culture, with all ages consuming. The popularity of manga led to popularity of anime, opposite of how the popularity of the two forms grew in America. It is sometimes hard to fully understand this relationship for American cartoon shows, while based on comics, hardly follow them. They use the same characters and general story arcs, but try developing their own identity.
Anime tends to be based directly on the source manga. Some changes are made in personalities or minor story points, as in Death Note, but otherwise try to be faithful. Try is the optimal word, because turning a manga into anime faces complications. Anime adaptations normally start when the manga is being published, to capitalize on popularity. Filler episodes or filler seasons are used, allowing the mangaka (writer/artist) time to develop story arcs.
Bleach and Naruto are known for having whole seasons of original content. Dragon Ball Z has fights lasting ten episodes while in the manga they are one or two chapters. These are special cases, with the anime being popular enough to sustain themselves through countless fillers. Others were not so lucky. Full Metal Alchemist and Hellsing both initially followed the manga, but created haphazard endings when the anime caught up with the manga. Soul Eater and Pandora Hearts had to throw together endings when the series were prematurely cancelled. These endings make almost no sense and ruin good anime.
The manga stories work. They contain the mangaka’s vision and present tight-knit stories requiring no changes. In recent years, anime makers realized how great the source material is, giving many series a second take. The 291 episodes of Dragon Ball Z were edited down to 159 episodes in Dragon Ball Z Kai, removing all filler episodes and elongated fights.
Hellsing was remade as Hellsing Ultimate, a series of OVA’s directly mirroring the manga. The same happened when Full Metal Alchemist got its second series Full Metal Alchemist: Brotherhood. This series does suffer in the beginning for episodes were rushed. They are in the first anime, and producers did not want to repeat. The result is episodes a little hard to watch, with the rest of the anime being the best I have seen. Recently the remake of Sailor Moon, Sailor Moon Crystal, finished airing, with each episode a direct translation of one manga chapter.
Not all animes are remade. Inuyasha was put on hiatus for three years, allowing the manga to finish. The result was The Final Act, the best arc in the anime. Bleach is currently on hiatus, waiting for the final arc of the manga to conclude before beginning production. One of the more interesting cases is Black Butler. Episodes in season one sporadically follow the manga, while the rest and season two are original. Instead of remaking, the original stories were ignored, with Book of Circus using flashbacks to reference the preceeding manga story arcs. The latest story arc, Book of Murder, was released as two feature-length films instead of the traditional series.
The manga stories do not need changing, and make better anime series. I wonder what they will remake next.