Spider-Man saving a little boy from a group of bullies, then fixing his science project, THEN offering to walk the kid home. It's nice to know that Peter, who we can safely say has had enough experience with being bullied, finds stopping this kind of thing important enough to turn up as Spider-Man.
Peter: Hey, Spider-Man. Thanks for stepping up for me.
And also, the crowd's reaction to Spider-Man coming back.
Hell, New York's reaction to Spider-Man in general. Compared to the Raimi films Spidey is beloved by the city, its people, and workers instead of harassed. It works both ways, as well, as Spider-Man here is quite a bit more direct in his interactions with the citizens of New York, a prime example being working directly with the firefighters to hose down a frantic Electro (Spidey even donning a firefighter helmet) and recognizing their efforts with a few friendly high-fives and throwing some compliments. Addressing them by name each, mind you.
Spider-Man offering to take an electrically charged Max away so the two can talk. If that damn sniper hadn't taken the shot, there probably would have been two heroes protecting Manhattan.
Honestly, Spidey's entire handling of the situation was this. Whereas many other incarnations would likely bring out the fisticuffs first and foremost, and only try to talk things out upon realizing they made a mistake, Peter in this iteration recognizes immediately that poor Max is confused and scared. Up until the very end and he has no other choice, he's trying desperately to talk things out with him, proving that Max's hero was pretty much everything he'd hoped he'd be. Unfortunately, the police (and civilians, for that matter) opted to act with nothing but disdain, completely undoing what Spider-Man was trying to achieve. As the above states, with Max's prior admiration, it's very likely he could've used his gifts to protect the city.
On the villain's side, Harry telling Electro he needs his help, and Electro joining him because this is probably the only time he has ever felt truly needed.
In addition to this, they both fill out their part of the bargain. Unlike most examples of Villain Team-Up, neither of them ever tries to double-cross the other.
Harry also has a very heartwarming reletionship with his assisant, Felicia Hardy. The way Felicia keeps asking if he's alright when he starts feeling sick and the fact that he asked her name because he wanted to be friends. And add to the fact that she tells about the off-book projects because she thinks that it will help him. In a deleted scene, Felicia come across Harry as the Green Goblin and intead of killing her, even though she was indirectly responsible for his appearance, spares her life.
Speaking of villains, Norman Osborn explaining his treatment of his son and handing over his life's research on the disease killing their family line. Even if the theories of him surviving were true, a cure for his stage of the disease seems extremely unlikely, and he had plenty of qualified scientists to leave it to. The fact that he filled Harry's childhood with an education in this work and personally handed him the data made it feel like a Take Up My Sword moment.
As a meta example, the filmmakers decide that instead of putting in the scene of Harry telling Peter he's being monitored, we get them awkwardly reuniting after so many years apart then slowly but surely reconnect as old but best friends. Makes watching what happens later all the more painful.
Also bear in mind that the filmmakers have the ridiculous task of setting up Harry as Peter's BFF even through the first movie completely ignored all of that. Then the Snark-to-Snark Combat starts, and we learn that Peter and Harry may not be two peas in a pod, but at least they're a pea and a bean in a pod.
Spider-Man spelling out "I Love You" in spider-webs for Gwen and then their interaction on the bridge. He tells her that they are not going down separate paths - that she is his path. And tells her that he'll go with her to England. Which doesn't ever work out - and makes her death hit that much harder.
When Gwen falls and Peter goes after her to try to save her, he shoots out his webbing and for a moment the webbing takes the form of a hand, reaching out to her one last time, making this moment heartwarming and a tearjerker at the same time.
The scene where Harry and Peter talk is a very touching scene, as it shows their friendship very well. And then Harry gets injected...
Aunt May's talk with Peter when he tells her that he wants to know more about his parents. She tearfully tells Peter that she resents Richard and Mary for just abandoning their son, and that she is the one who raised him and took care of him.
"You're MY boy! Mine."
And Peter responds by telling her that he genuinely appreciates everything she's done for him over the years and that she won't be replaced to him, and comforts her when she starts crying.
In Brazilian dub, this scene also counts as a Meta Heartwarming, as the voice actress who dubs Aunt May is the real-life mother of Peter's voice actor.
When Gwen says a simple, "Goodbye, Max," when she gets off the elevator. Max is left surprised and happy that she actually remembered his name. Doubles as a Tear Jerker since it shows just how bad a life he's had.
From the video game version of the story, the very ending. Peter Parker gets a pep talk about the will of his late uncle from none other than Stan Lee himself, who tells him that Uncle Ben was always proud of the man that he was becoming...and that he shouldn't focus on how his uncle died, but how he lived. Finally, Stan Lee tells Peter to go out there and be the hero that Uncle Ben always knew that he'd be...and Peter agrees without hesitation.
Max becoming Spider-Man's fan, defending him on the radio, and even daydreaming aloud about Spidey remembering his birthday. It's certainly pretty obsessive and weird, but it's also kind of sweet and childlike.
Peter finding out that his father didn't run off with the research for the money, but because it was the right thing to do - even though he and his wife had to leave their son behind, with the last line being about how much he loves Peter.
The deleted scene featuring Flash Thompson. Flash happily walks up to Gwen & Peter, and proceeds to tell them that he's got into college before thanking Gwen for the tutoring sessions, before hugging Peter. Whilst it's only a minute or two long, it shows that Flash continued to shed the Jerk Jock attitude between films.