Spider-Man’s a superhero whose central theme is conveniently spelled out for us: with great power comes great responsibility. And it’s a great one too. A nobody gets given amazing powers and has to learn what to do with them. It's a pretty essential part of most incarnations of Spider-Man, be it Miles Morales or even more recently when it's Gwen Stacy that gets bitten by the radioactive spider and becomes Spider-Woman. It's always that balance of power and knowing what to do with it.
When there comes time for a cinematic Spider-Man that's the theme of the (two) hour(s). In Sam Raimi's original film, Peter Parker's irresponsibility is what gets Uncle Ben killed, and his acceptance of his responsibility leads to him fighting Green Goblin. The conflict of the second Spider-Man is him giving up the mask, only to take it back up because he's the only one who can stop Doc Ock. In Marc Webb's Amazing Spider-Man we see Uncle Ben die (again), providing the impetus for Peter to use his powers to stop crime. Powers, responsibility, and Peter Parker reluctantly being the hero.
So Spider-Man: Homecoming seems to have its theme waiting for it: responsibility and all that (most likely through the death of Uncle Ben). Except Peter is already Spider-Man. And Uncle Ben is already dead. And Peter really likes being Spider-Man.
Right here this sets up a different sort of superhero narrative. The usual internal conflict for a superhero is their unwillingness to do the heroing (and so the climax is them deciding to hero). Tony Stark becomes Iron Man out of a sense of guilty responsibility, not for the fun of it. Thor’s a self-serving blowhard who learns humility. Batman operates out of a just vengeance. Spider-Man usually Spider-Mans out of a sense of responsibility. But this Spider-Man really likes crimefighting; he gets a thrill out of the heroics. In fact, he wants more: he wants to be an Avenger. Like Iron Man.
It's hard to give an eager hero internal obstacles. Tony Stark is hung up on guilt and the idea that he has to do it alone which makes things difficult for him. The Guardians have to overcome their infighting and greed to fight Ronan. Even Captain America questions if it's worth it. But Homecoming's Peter is go-go-go. He's got the power, and he's fighting crime with it. Where's the classic Spider-Man theme?*
Here's part of Homecoming's genius. Responsibility in this movie doesn't just mean crimefighting and heroing, it's the reason for doing so. Peter, in the aftermath of taking part in Civil War's airport battle, wants to be an Avenger. He wants in on the big leagues. He bugs Happy Hogan to tell Tony about what he's doing and he chases the Vulture because this is his chance to make it big.
The film's climax, and Peter's self-actualization, comes when Peter decides to hero not for the glory or to impress Tony, but instead to save the day. It may not sound like a huge difference, but, without spoiling anything, the film makes the distinction clear. It’s when Peter heroes for the greater good and not for himself, that he becomes a real hero. Spider-Man Homecoming is still a movie where Spider-Man learns a lesson in responsibility, it just plays out differently than usual.
We've had enough versions of Spider-Man over the past fifteen-odd years for the idea of a new Spider-Man to be met with a hint of tiredness. Here we go again, Spider-Man has to learn how to Spider-Man and responsibility. And Homecoming is about that, but it handles it in a much different manner than prior renditions. You don't need an edgy and avant garde narrative with brand new everythings to tell a new story. Sometimes just digging into your core theme is enough. I think that's why Homecoming is able to be quintessentially Spider-Man while still feeling incredibly refreshing. Jon Watts and the team didn't feel the need to completely reinvent Spider-Man, rather they explored the story a bit more and found something new.