There are a lot of differences between the Death Note Manga and it’s Anime adaptation, but they’re mostly minor differences, with the overall story being the same. Compared to the Manga, the Anime has had some scenes removed, new ones added, and some included scenes have been altered. I’ll go over some of the more notable changes, but this won’t be a complete list.
The conversation between L and Light that happens in the coffee shop was altered significantly, with the Manga scene being a lot longer and containing more dialogue between Light and L. In the Anime the only question L asks Light is the one where L gives Light pictures of the “L, do you know that Gods of Death love apples?” messages. In the anime L also asks Light what he can conclude by L revealing himself, and also asks him a question about the various FBI detectives that were killed by Kira.
The following scene, with Light, L, and Soichiro, had alterations as well.In the manga L asks Light what he thinks Kiras personality is like, which leads L to state that the most likely person to be Kira based on Lights description would be Lights sister, Sayu. This causes Light to be outraged, and Soichiro to state that he doesn’t believe it’s possible for Sayu to be Kira (but doesn’t make the same statement about Light, which is something Ryuk makes note of).
In the Manga the first meeting between Misa and Light contains an explanation from Misa about how much of a persons face she needs to see in order to know their name, which was omitted from the Anime.
The anime also excludes a few scenes between Rem and Higuchi, likely to prevent the watcher from discovering who the new Kira is too long before it’s revealed (due to recognizing the voice). The most notable exclusion is a scene where Rem tells Higuchi he can stop killing criminals now, but Higuchi refuses, stating he still needs criminals to die, since it’s good for business.
The manga also shows that Rem is disgusted by Higuchi and the other members of the “Kira board”. Rem tells Misa during their bathroom scene that her disgust at the actions of the “Kira Board” have made her come to the conclusion that Light may actually be a good person, and wishes for him to succeed (and, for this reason, is no longer able to kill people in order to help Light succeed, since it would cause Rem to die).
When Light gets the notebook back, and Misa resumes her duties as the second Kira, there’s a scene in the Anime which shows Misa walking in the city singing. This scene isn’t present in the Manga, and was added to the Anime.
The death of L is handled differently as well. In the Anime there’s a scene which takes place on the roof of the task forces HQ, after which L gives Light a foot massage. Immediately after this scene L and Watari are killed by Rem. In the Manga the rooftop scene and foot massage are completely omitted. The scene before and after this, showing that Rem is considering killing L to save Misa, is all just one scene. It’s not broken up by anything in the middle. When L does die, he’s complete silent in the Anime, but in the Manga he says “I wasn’t wrong”. Finally, there’s a few scenes between L and Watari in the anime (flashback to L at the orphanage and L coming to Wataris computer room at HQ) which aren't present in the Manga.
The second half of Death Note, featuring Near and Mello, contains a lot of alterations as well, with quite a few scenes being removed. It also adds a few additional scenes between Misa and Light, giving a bit more of a glimpse into their relationship, and expanding some more on Misas character (you really get to see how much she cares for Light, and how concerned she gets when Light is upset).
In the Manga Near brings Soichiro in for questioning after Near sends a chopper to pick up Soichiro in the desert. The SPK also gets the USA president to send in a squad of soldiers to retrieve the Death Note from Mello, but the assault fails due to Sidoh taking off everyones helmets so Mellos group can kill them with the Death Note.
In the Manga Mello and Halle Lidner are shown to have met prior to the scene where Mello puts a gun to her head, and almost immediately after the meeting between Mello and Near, Mello contacts Mogi and sends him to meet near. Near asks him a few questions and shares some information with him, with Mello and the task force listening in (although Mogi doesn’t say anything). Mogi is actually still with the SPK when Demegawa storms their headquarters, and safely escapes with them. Mogi is still with the SPK up until Aizawa goes to meet up with Near, at which point Mogi and Aizawa leave together. In the Anime Mello never contacts Mogi, and Mello and Halle weren’t shown to be in contact prior to Mello putting a gun to her head. When Aizawa gets in touch with Near regarding his doubts about Light he does it over the phone, and not in person. Interestingly enough, though, the Anime scene where Aizawa does first meet Near in person was actually over the phone in the Manga, so they (sort of) swapped those two scenes.
The anime removes a lot of scenes between Misa giving up the Death Note and Light making contact with Mikami as well. In the Manga we actually see Light looking for someone to replace Misa, and notices Mikami during a broadcast of Kiras Kingdom, which results in Light sending Mikami the Death Note. Mikami also tries to reach out to Kira (Light) during an episode of Kiras kingdom, stating that if Kira can’t get in touch with him he’ll begin acting on his own. Mikami also doesn’t get in touch with Light until the second meeting between Takada and Light. In their first meeting Light gets Takada to make some statements during the broadcast in an attempt to have Mikami contact her, and then Mikami contacts her during the second meeting between Light and Takada.
In the manga we see a few more scenes of Mello and Matt (the one who helped Mello kidnapped takada) observing Misa and Light, but nothing really notable happens in these scenes.
As the other answer already made note of, the ending is changed as well, and is definitely the biggest change. In the anime Light escapes from the warehouse after Mikami kills himself, and dies when Ryuk writes Lights name in the Death Note.
In the Manga Mikami doesn’t kill himself in the warehouse, and Light doesn’t escape. Instead, after Matsuda uses Light for target practice, Light crawls on over to Ryuk and begs him to write everyone’s name down in his Death Note. Instead Ryuk writes Lights name in the Death Note, resulting in Lights Death. The story then cuts ahead a few years, and there’s an epilogue. In the epilogue we learned that Mikami killed himself in prison, and that Aiziwa has taken over as head of the Japanese Police. Near has taken over the position as L, and still works with the police. The final shot of the Manga shows that people still worship Kira, and pray for him to return one day.
The Anime also contains a lot more imagery than the Manga does. In the anime there are several scenes which often don’t show what’s actually happening when some characters are talking. For example, during the “I am in Japan” conversation between Near and Light it shows the two of them going up a construction elevator, whereas in the Manga it just shows them talking over the phone. These changes were likely made to make the long dialogue scenes of the show more visually interesting.
So that should cover the majority of the changes between the Manga and Anime, although I may have overlooked something. As you can see, the overall story remains the same, but there’s definitely enough differences between the two make reading the Manga worthwhile if you’ve already seen the anime (and vica versa).