Though many anime derive from manga, not every one of those anime follows its source manga closely. When it comes to anime that are different from the manga, plenty of examples come up. Differences between manga and anime versions can arise for all sorts of reasons: an anime geared towards kids may omit the violence seen in the manga or a studio may need to create filler episodes while awaiting the next manga chapter. You may not realize it, but Pokémon's anime and manga versions differ wildly.
Fans often bristle when their favorite manga aren't replicated faithfully. Imagine waiting for your favorite scene from the manga only to find it's substantially changed or entirely cut from the anime version. On the other hand, such differences mean each version can tell a vastly different story. Just look at the details left out of Akira from manga to anime. Occasionally, the original manga falls short of the anime, which can shift up the plot with exciting and unique filler sagas.
The 2003 version of Fullmetal Alchemist quickly goes off in a different direction from the source manga, a shift that led to the 2009 reboot Fullmetal Alchemist: Brotherhood. While the 2003 version retains many of the main characters from the manga, Fullmetal Alchemist features a new antagonist named Dante, a different method for creating Homunculi, and an ending that drops the protagonist into the middle of World War I.
Though a great story, the original Fullmetal anime is not even remotely the same as the manga - if you want to watch that story unfold on-screen, you'll find it in Brotherhood.
When the first Hellsing manga was coming out, there wasn't a lot of manga material with which to work. This meant that the producers needed to come up with their own villain, a particularly powerful vampire named Incognito. Though the war-hungry Nazis of the manga briefly appear, they take a backseat to Incognito's attacks on the Hellsing organization.
Hellsing Ultimate, a set of OVAs which began airing five years after the initial series' 2001 release date, followed the then-completed manga more exactingly.
The first season of Black Butler is a fairly straightforward adaptation of its source material, though many of the episodes do involve filler. The big changes come in the second season, when new characters who never appear in the manga dominate the storyline. Alois Trancy and his demon butler Claude Faustus appear; Claude and Sebastian Michaelis compete for Ciel Phantomhive's soul.
Most of this material is unrelated to the storyline of the first season, which focused on Ciel's powerful need for revenge against the people who killed his family. A 10-episode OVA, Book of Circus, ignores the second season, instead working straight from the manga.
The first season of Tokyo Ghoul sticks more or less to the manga, but that changes dramatically during the second season, Tokyo Ghoul √A. In the manga, Ken Kaneki works against Aogiri Tree, an organization attempting to bring liberation to ghouls through violent means, including kidnapping and terrorism. In the anime, Kaneki takes the opposite approach, joining Aogiri Tree instead.
On one hand, Tokyo Ghoul √A presents a fresh storyline for Kaneki; on the other, it's frustrating not to see the manga's storyline adapted. In addition to tweaking Kaneki's story, the Tokyo Ghoul anime also removes many of the more violent scenes seen in the manga.