There are seven books in the original Harry Potter series and there are eight movies in the film adaptations. Despite the fact that the movies outnumber the books, there are still a ton of moments from the books that were left out of the movies for no reason at all.
After all, in the Harry Potter books, there was way more time that the reader could spend in the world with the characters than in the movies. While some cuts were necessary and others were inadvertent, there were many that had no reason to be removed.
S.P.E.W.
The biggest arc in the books that was removed from the movies is the story of S.P.E.W., Hermione's seemingly-fruitless effort to liberate the house elves of the wizarding world.
The books do an excellent job of showing how hellbent on justice Hermione was for all creatures, and the loss of that in the later movies takes away a bit from the narrative that the Trio is fighting for everyone and not just those who look like themselves.
Ron As A Quidditch Hero
While S.P.E.W. is a big part of Hermione's arc, a lot of the elements of Ron's character development are shoved aside in favor of the overarching story. For example, in Harry Potter and the Order of the Phoenix, Ron's ability to win quidditch for Gryffindor in spite of Harry's absence is a big moment in his self-actualization.
Instead, the fifth movie cut way down on Ron moments, in favor of the sprawling narrative at play. It's enough to make fans wish that the massive length of the early films was still present.
Weasleys' Wizard Wheezes
Speaking of the Weasleys, there's a crucial moment between Harry, Fred, and George that is omitted entirely from the movies. While S.P.E.W. and quidditch are big arcs, this moment would have taken up maybe half a minute of screen time.
After winning the Triwizard Tournament, Harry gives his winnings to Fred and George, which they use to open up Weasleys' Wizard Wheezes, their joke shop. In the movies, Harry never gives them the money, so the shop just opens randomly. There was no reason not to include a little moment between Harry and the twins!
Peeves
Speaking of wizarding pranksters, the ghost of Peeves was left out of the Harry Potter films. However, every single one of his moments in the books would have been perfect on screen!
Rik Mayall was cast as Peeves and then later cut from the first film. However, Harry's relationship with mortality and with ghosts is a pivotal, emotional one in the books. They don't get nearly enough play in the films and Peeves would have been a great way to introduce them better.
The Muggle Prime Minister
At the beginning of Harry Potter and the Half-Blood Prince, the Muggle Prime Minister is given an actual role in the story for the first time. He was previously only mentioned in passing.
It's an interesting way to ground the stakes of the story on a global scale and not just in the wizarding community. Scenes between Kingsley, Fudge, and the Muggle Prime Minister are an interesting lens to view the events of the story. It could've made for a great prologue in the films, but instead, it was cut.
"Remember My Last"
One of the best moments in the entire book series is when Dumbledore sends a Howler to Aunt Petunia at the beginning of Order of the Phoenix. She's ready to kick Harry out of the home on Privet Drive when Dumbledore intervenes, and she silently goes back on her intent.
It's a brief moment, but it's so full of awe and so telling about how Petunia cannot completely rebuke her ties to Lily. Unfortunately, the movie removes it entirely, and for no good reason. It would've been so amazing to deepen the story, which was already growing darker. Instead, it was chopped away.
Peter Pettigrew
If fans have only ever seen the movies, they might be confused about what happens to Peter Pettigrew. The Marauder who betrays his friends, Peter is killed in the books by his own hand, after recalling the life debt he owes to Harry.
In the movies, Pettigrew pops up at Malfoy Manor and is completely absent from the second part of Deathly Hallows. Book readers know that he was killed, but movie-goers might be curious about that seemingly unresolved character thread. He is knocked unconscious with a brief "Ow," and it's not entirely clear if he is dead from the moment.
Neville's Parents
The movies rightly place an emphasis on Harry, Ron, and Hermione, but it's often done at the loss of other characters and their nuances. Neville gets the shaft in the middle films, with an emphasis on his persona in the early and later movies.
This misses out on a ton of what Neville brings to the story. His parents, who lost their memory and sanity after endless torture in the first wizarding war, make for a powerful moment in the books. They're left out of the movies and only mentioned in passing, for no reason other than to hustle the story along.
Horcruxes
The production design in the Harry Potter films was incredible, but they left out a number of specific props mentioned in the books for no good reason. Some of these props (like a locket or a diadem) come back as horcruxes, and the movies suffer a bit from not having previously introduced them.
They're clearly mentioned in the early goings of the story. With the Room of Requirement so full of objects as it is, it wouldn't have hurt to include a few of the named items.
Lupin's Fear
Lupin is a major character in the films. At first. By the end, he slides into the default mentor role after Dumbledore's passing, and much of his arc is dispatched in favor of following the Trio on their adventures.
This causes the scene where Lupin tries to run away from his family and gets yelled at by Harry to be cut from the films. There was plenty of time to include it, but the producers just opted not to. It takes away from his complexity. That's what the books will always be for.