While there are many anime that don't necessarily follow their manga, most of the time the anime that do this end up way worse. For example, the original Fullmetal Alchemist and Fruits Basket had super bad and rushed endings that either didn't really tell the whole story or were left super open-ended that they both ended up receiving new anime to retell the story closer to the manga's original story.
Other anime that don't often follow the manga seem to fall weak, such as Tokyo Ghoul, which diverges completely and feels like an entirely new show. However, there are still a few anime out there that have better endings, that are often totally original, compared to their manga.
10 Space Dandy
The Space Dandy anime ending is way better than that of the manga as the manga came after and was made by Square Enix. The manga is rather short and just follows Dandy and the Aloha Oe crew through a few more space adventures. The anime doesn't really end as the series as a whole is very open-ended with no real continuality, but it was still extremely exciting and original. The anime was way better than the manga in terms of animation and perfectly represented Dandy's flashy personality.
9 K-On!
The K-On! anime is way more elaborate than the manga and is more than just a short little manga. The anime actually goes deep into the story and characters, not to mention having a super nice soundtrack that is fitting with the band's music. The K-On! manga runs more like small little clips that happen between the characters and is more centered on the comedy, while the show is actually fleshed out and a full-fledged anime. The ending is also way better.
8 Death Note
While the anime continues mostly in the same way, the way Light Yagami dies in both stories differ and is ultimately better in the anime than in the manga. In the manga, Light begs Ryuk to kill Near and Matsura, the detective that's on to Light, but Ryuk tricks him and write Light's name instead. The reasoning was that Ryuk didn't want to wait around for Light to die, and it would be even worse if Light was in jail and Ryuk had to wait by.
In the anime, Light gets shot by Matsura as he tries to use a piece of the Death Note to kill Near. He goes out on the run, bleeding as the cops are hot on his tail. Ryuk writes Light's name in the notebook, acknowledging that Light's death was inevitable.
7 Yu Yu Hakusho
While the manga ending is widely adored, it is often thought of by fans of Yu Yu Hakusho that the final arc of the anime is done way better than in the manga and is way more rounded out, though it was still not widely loved. It is possible that the fear of ending Yu Yu Hakusho poorly may be settling in for Yoshihiro Togashi and could be a reason he has yet to complete Hunter X Hunter. However, the fact remains that the anime was definitely done better than the manga and shows the characters each a final time.
6 Bunny Drop
Bunny Drop doesn't necessarily have a different ending from the manga, it just ends at a much earlier point before things start to get weird. In Usagi Drop, 30-year-old Daikichi becomes the legal guardian to his grandfather's illegitimate six-year-old daughter (which weirdly makes her his aunt), Rin. Daikichi and Rin have an adorable relationship as the series continues as it shows this young and new dad struggling to raise the young girl. The anime ends while Rin is still a child and has a good relationship with Daikichi. In the manga, however, it continues and skips ahead by 10 years, making Rin 16 and Daikichi 40. Things take a sharp turn when Rin starts to develop feelings for Daikichi.
5 Tsurezure Children
Tsurezure Children follows a bunch of different love stories between different students. Many fans of the series adore the anime adaptation due to the fact that the show stays with the characters for much longer than in the manga. While there is no real ending for either the anime or manga, the stories of the characters live on and are better told throughout the animated adaptation compared to the manga. Characters are seen more frequently and the short moments are explored better.
4 Cowboy Bebop
The Cowboy Bebop anime actually came before the manga and has a way better ending than its adaptation. The ending of the anime is utterly iconic and emotional as Spike fights Vicious and loses his life. It is an iconic scene that has made anime history.
But the manga doesn't have anything of the sort. The Cowboy Bebop manga is short one-off chapters that show the Bebop crew going on more adventures together throughout space as if they happened between episodes of the anime.
3 Shaman King
The ending of Shaman King differs between the anime and manga in one crucial way, and that is the death and forgiving of Hao. In both iterations, Hao becomes the main antagonist and ends up fighting against the main characters, bent on destroying humanity once he becomes the Shaman King. In the manga, they defeat him and give him another opportunity to redeem himself, but in the anime they kill him. His death is tragic and makes for a seemingly good ending.
2 Hunter X Hunter
Though Hunter X Hunter has yet to complete, anime-only fans are having a much easier time adjusting to where the series left off rather than where the manga has seemingly gone on hiatus. While there are still possibilities that both the anime and the manga can come back eventually and finish the story, the anime is a lot more rounded off and has a more concise ending that won't leave viewers begging for more like the manga.
1 Hellsing Ultimate
While the original Hellsing had a pretty bad ending that heavily deviated from the manga, Hellsing Ultimate tried to stay much truer to form than the previous series. Both endings are rather similar, but something about the way it unfolds throughout the anime is done so much better. In the series, the gore is way more fitting and accurately displays the intensity of the scene and makes the entire ending feel a lot more complete than in the manga.