What's the difference between the Full Moon wo Sagashite manga and anime?

Question

I read a lot of the Full Moon wo Sagashite manga a few years ago and recently started watching the anime. It's been so long since I read the manga that I can't tell how much is different, but I remember my friend telling me a few years ago that they are very different.

Is there a lot of filler in the anime? Is the plot/story significantly different?

Please mark any spoilers as corresponding to either the manga or anime so those who have only seen one or the other don't get spoiled if they don't want to.

(If my friend was blatantly wrong and they are pretty darn close, I apologize and don't mind having the question closed.)

Answer 1

Here are the differences, there might be some spoilers. (If you know the story and you think something is a spoiler, please let me know in a comment so I can fix it).

1. Anime: Mitsuki claims to be called FullMoon during the audition.
   Manga: This happens in another moment.

2. In the Manga, Meroko's past is revealed: When alive her name was Moe and she was Mitsuki's grandma's best friend. She killed herself believing her friend had accepted the marriage proposal by the guy she was in love with.
   In the Anime, nothing is revealed except something about her Shinigami past, next to Izumi whom she loved.

3. In the Manga, also Izumi Rio's past is revealed: When alive his name was Lio. He had a hard childhood. His father died when he was still very young, while his mother went crazy for the suffering and started to mistreat him, despising him and accusing him to be the reason why her new man left her. For this reason, in order to see his mother smile again, he kills himself being run over by a train.
   Farther in the Manga, we discover that Izumi's vision of his past is slightly wrong...

4. Manga: Fullmoon single is "Angel".
   Anime: It's "Myself" by Myco (singer and Fullmoon's voice).

5. Fullmoon's second song Eternal Snow (always sung by Myco in the Anime only). The history of the carillon, that contains the song's melody, written by Aoi for Mitsuki and that the girl finds in a closet outside of the house (where her grandma locks her for punishment), is told by Takuto when he regains his memory.

6. Anime: Another difference about Eternal Snow, appeared in the anime only, is the challenge with Madoka who, after a snitch, decides to make her debut too using the same song. Mitsuki feels sad about this since the song was written by her father and so it belonged to the Route L. But Takasu-san, the producer, refuses to produce it and worse, decides to produce Madoka in its place. So Ooshige and Fullmoon end up being without a producer. Ooshige is however a Route L's fan and so he askes Wakaouji to produce this song. He initially refuses, because he swore to stop having any relation with music. In the Anime Mitsuki runs away but she comes back after a day.
   Manga: Something similar happens in the manga, where a song, written by Mitsuki is then found and sung by Madoka because of Izumi. The song's name is "Princess Marmeid". Furthermore, Wakaouji is not contacted by Ooshige, rather he spontaneously proposes himself as Mitsuki's producer, since he was searching her that ran away from her house. In the manga she doesn't come back until almost the end.

7. Manga: Takuto Kira, the Shinigami, "dies" jumping from the hospital's roof, seeing by Wakaouji. Takuto Kira received a surgery to his vocal chors, because of a cancer. They are removed and since he lost his voice and cannot sing anymore, this brings him to kill himself.
   Anime: When the throats problem is found out, Takuto rides a bike as fast as possible, deciding not to turn at a bend he encounters.

8. Manga: When Takuto was alive, he had a girlfriend named Hikari, a beautiful model, that will meet Mitsuki too.

9. Anime: Meroko is promoted from Shinigami to Angel.

10. Another big difference is about Eichi-kun (huge spoilers):

    He was Mitsuki's first love.
    Anime: He dies two months after he arrived to America. Mitsuki knows nothing about his death and will find out during her trip to America, when she decides to visit him along with Ooshige, Wakaouji, Takuto and Meroko.
    Manga: She knows about his death but she refuses it, so she talks as if he was still alive. This reactions sprouts from her feeling guilty because of the lack of reaction when he confessed his love for her, after kissing her. In the manga, this guilty feeling is much stronger and dramatic than the anime's, this is because she runs away after they kiss, the day before he went to America, while in the anime the kiss is not shown.

There are more differences in the Wikipedia page. If you want I can include them.