Disclaimer: This can be a place to discuss Spider-Man across multiple mediums. I have not seen The Amazing Spider-Man yet, although now that I've gotten some distance from all the controversy I think I can give it a chance.
Spider-Man is a character and concept full of complications. He's a street level vigilante that is responsible, however indirectly, for most of the villains he fights. He's brilliant without profit. Being a super hero is not directly tied to his brilliance (re. webshooters replaced with organics with little story modification). He's also, simultaneously, portrayed as an every man. He's racked by guilt but uses humor as a defense mechanism. Ironically, even after the life lesson learned from the death of his Uncle Ben, he's still opportunistic in how he takes pictures of himself in action and in recent comics invents at Horizon Labs using knowledge he's cribbed from adventures as Spider-Man.
Compare Peter Parker/Spider-Man to Tony Stark/Iron Man and Bruce Wayne/Batman. Stark and Wayne use their brilliance in crime fighting, but also run benevolent corporations that, to a certain extent, are more effective than or at least complimentary to the crime fighting. Parker, arguably, accomplishes little as Spider-Man. One man patrolling an entire city may stop the occasional mugging or bank robbery, but what he lacks is presence. Through their very existence, Iron Man and Batman create peace. Iron Man is basically a walking nuclear bomb that has "democratized peace", while Batman keeps Gotham in a perpetual state of fear. Spider-Man doesn't reign over New York with fear, so what is he really accomplishing?
This is an inherent problem with most superheroes. Do they cause more problems than they stop? What Spider-Man lacks in most incarnations, and specifically in the Raimi Spiderman series, is introspection. The Iron Man films have explicitly covered the danger of the spread of Stark technology and Iron Man's responsibility over that technology. Nolan's opinion that Batman is actually hurtful to Gotham is evident in his trilogy, and the sense of escalation is a main theme of the series.
So what does Spider-Man mean to New York? Oddly enough, Maguire's take on the character doesn't come across as a boy from the big city at all. The suburbs of Queens might as well be a small midwestern town for all of Parker's lack of world weariness. What Spider-Man can be for New York, however, is the epitome of good Samaritan. He may not curtail a crime wave, but what Raimi nails and Maguire personifies is naive but uncynical good intention. Spider-Man isn't at war with crime, he's just doing his part. That's why the bridge scene, and of course the elevated train scene, in Raimi's movies work perfectly for me.
The way he changes the city is by his influence. His oft-quoted motto, as well, isn't an abstract concept about fighting crime because you can. The reason that the villains are always directly connected to Parker is to offer up concrete examples of taking responsibility for your actions. This is also fascinating considering Raimi's known penchant for abusing his actors and characters, and his having been raised in Conservative Judaism. Much like the very Jewish running theme through the Coen bros. movies of actions on Earth receiving repercussions, Spider-Man's guilt and past mistakes manifest in Old Testament ways. Much like the strong judgment in Drag Me To Hell, the super villains in Raimi's Spiderman series appear whenever Spider-Man acts selfishly.
The Green Goblin exists, yes, almost from the start of the first movie but only makes trouble in Parker's life when he acts selfishly. That is, when he encounters Mary Jane again in Manhattan and when he takes the freelance photography job at the Bugle. Raimi seems to be perfectly aware of the strange contradiction of Parker taking advantage of his own infamy in that way. The second movie sees Doc Ock manifest when Parker sets Spider-Man aside to focus on his classes. The third movie is the most blatant with this cause/effect correlation, as New Goblin shows up when Parker plans to propose and Sandman when Spider-Man wins the key to the city.
In that way, Raimi's Spider-Man is a New Yorker in how he follows in the tradition of NY Jews like Woody Allen: he can't catch a break. This, however, is inherent with its own contradictions as Raimi loves Christian and specifically Catholic imagery: Spider-Man in the Christ pose in the second movie, the Cathedral in the third movie.
What's important for a Spider-Man story, and what Raimi understood, is the tone. It only works with a heightened, magical realism feel. Spider-Man is an inherently silly concept, and doesn't lend himself to philosophical musings about world peace or ruling through fear. He has personal demons that manifest as villains, not unlike Batman, but unlike Batman these villains exist to punish Spider-Man for the slightest deviation from the path. This may seem incredibly harsh, and comparable not just to Drag Me To Hell but the Coens' A Simple Man, so the only way to present it without coming across as nihilistic is with a bright brush. Bright colors, spectacle, fun, and hope.
What sets Raimi's Spiderman movies apart from his other films is the protagonist may be put through the ringer but he never loses his hope.
In conclusion, I plan to see The Amazing Spider-Man soon for a good comparison. We'll see...