Is Web-Shooting A Built-In Superpower? Or Does Peter Make His Own Shooters?
The first Spider-Man movie seems to be checking off "Classic Spider-Man Requirement" boxes at first. He's a dork, he's friends with Harry Osborn, he has a crush on Mary Jane Watson, and he fights a pro wrestler, because apparently that's a real sport in that universe. But then he shoots webbing, not out of some wrist gauntlets, but out of his own gross wrist flesh. If you're new to the character, it makes sense. If a souped-up spider bit you and gave you all sorts of new abilities, you'd be disappointed if web-spinning wasn't one of them. That's like 90% of what a spider does!
But in the Amazing Spider-Man films and the MCU, Spider-Man has artificial web shooters, which is how it's almost always been in the comics. It's the story of a guy who gains the well-known spider abilities of wall climbing and magically sensing danger (if you ever see all of the spiders fleeing your house, follow them out), and then decides to lean into it by building some web shooters.
So why did the first film make the change? Well, before Sam Raimi was attached, James Cameron was gonna take a crack at it. And in his treatment, Spider-Man's webs were biological, and this is one of the only things from it that Raimi liked, because he felt that it made Peter Parker less of a relatable everyman if he was a teenage engineering prodigy along with being a powerful arachnid hero.