Harry Potter is, of course, the hero of the Wizarding World. He defeated Voldemort as a baby and a teen. He was a genius at Quidditch and Defense Against the Dark Arts. But, he was also a boy. There was magic involved, obviously, and the fate of the world, but Harry was also a boy and a teenager and did some very boy/teenager-y things, including being cruel to some of the people around him.
He wasn't only cruel to bullies who deserved it, like Draco Malfoy or Snape, but to his other classmates, and housemates, Neville Longbottom in particular. Neville thought Harry was his friend because poor Neville didn't have friends but really, the boys were nothing but acquaintances.
Harry Doesn't Help With Trevor the Toad
Trevor needs a lot of help from the very beginning. As soon as Neville boards the Hogwarts Express, his pet toad goes missing. Maybe we can't hold it against Harry for not helping a stranger find his misplaced pet, but Harry is still ignoring Neville's needs in book three, Prisoner of Azkaban. When Neville once again messes up in potions, Snape tells him they will be feeding a few drops of Nevill's failed potion to his beloved pet when class is over (which could have been a death sentence), and Harry does nothing about it.
Potions Class in General
Neville always struggled in potions class, even on days when the life of his pet was not threatened by the Professor. However, even as Neville's struggles continue, and, even after Neville's greatest fear, Snape, the Potions professor himself, is revealed to his classmates, Harry does not offer his assistance. Even if Harry was not confident in his own potions abilities, he doesn't even consider sitting with Neville in class, just as a good friend or as moral support. It's not a great look for the Boy Who Lived.
Leg Locker Curse
In year one, Harry Potter and the Sorcerer's Stone, Malfoy performs a leg locker curse on Neville, who is then forced to hop around much of the castle before anyone stops to help him. The person who finally helps him is Hermione. Ron and Harry only snicker, and Ron offers him advice on standing up to bullies. It may still be early in their relationship, but the boys share a dorm, and it's still as if the great Harry Potter looks down on Neville Longbottom.
Odd Man Out in the Dormitory
In Harry's year, there are five boys who share the dormitory. There is Harry Potter, Ron Weasley, Dean Thomas, Seamus Finnigan, and Neville Longbottom. Harry and Ron are best friends and Seamus and Dean are best friends, and this leaves Neville as the odd man out in his own dormitory.
Even after so many years of sharing a room, Harry never takes a special interest in Neville or invites him to join conversations between Ron and himself. He still leaves Neville to his own devices. He doesn't even realize Neville invited Ginny to the Yule Ball.
No More Dumbledore's Army
Despite the success of Dumbledore's Army in book/movie five, The Order of the Phoenix, Harry is unwilling to continue the club in book six, The Half-Blood Prince.
Despite Neville's request, and, despite the bonds that were made during the classes, it's not enough to Harry that people like Neville and Luna seemed to need and thrive in the environment. One would think if Neville and Harry were real friends, that Harry would stand up and agree to help Neville and make his time at Hogwarts as enjoyable as possible.
Talk About Their Parents
In the fifth book, Harry Potter and the Order of the Phoenix, Harry, Ron, and Hermione, learn that Neville's parents were tortured by Death Eaters until they lost their minds. While Harry had learned the information in the previous book, it wasn't until the trio ran into Neville at St. Mungo's that Neville was aware that anyone knew about his parents.
Even after the truth came out and Harry learned that he and Neville shared a traumatic history with Voldemort, his followers, and their loved ones, the pair never discuss it. They never connect on the fact that they both suffered greatly under Voldemort's original reign.
Petrificus Totalus
Sure, readers and viewers can all think it was necessary, but that doesn't stop it from being cruel. When Neville confronts Harry, Ron, and Hermione as they again sneak out of Gryffindor Tower at night, Hermione petrifies Neville as he threatens to fight them, to keep them from losing Gryffindor any more points.
While it was necessary for the trio to go after the sorcerer's stone, Harry never speaks up at all about Hermione's treatment of their fellow Gryffindor. One would think that if Harry had any friendly feelings towards Neville at all, he might have commented on it when the boy was petrified.
The Half Truth
In Harry Potter and the Deathly Hallows, Harry passes Neville as he walks toward the Forbidden Forest to confront Voldemort. He pulls off his cloak and tells Neville to kill Voldemort's snake, Nagini.
What he doesn't tell Neville is why he needs to kill Nagini. Perhaps you can say Harry was short on time, but it really seems like he was still holding information a little too close to the chest, only trusting Ron and Hermione with everything he knew. If Neville was Harry's real friend he should have been in on it all as well.
To the Department of Mysteries
In Harry Potter and the Order of the Phoenix, Ginny, Luna, and Neville insist on going with Harry, Ron, and Hermione to the Department of Mysteries after they all escaped from Umbridge. In the novel, Harry admits that these three, including Neville, would not have been the members of Dumbledore's Army he would have chosen to assist him on the mission. Not a very friend like thought to have.
Banned from Hogsmeade
In the third book, Harry Potter and the Prisoner of Azkaban, Harry is unable to visit Hogsmeade because he doesn't have a signed permission form. Neville is banned from visiting the village after he loses a list of Gryffindor tower passwords that allow Sirius Black into the castle. When Harry receives the Marauders Map from Fred and George, he sneaks into Hogsmeade alone. He doesn't ask Neville to join him, and, in fact, lies to Neville to get him to leave him alone so he can get away.