I don't think so, personally.
If there weren't a set of Spider-Man movies in existence (and well-regarded, for the most part), I don't think the ASM movies would still be all that popular. Why? Because of two fundamental things: story and character. The stories are no more than a series of set-ups for action scenes and they're just so far off the mark with the character of Peter that it's upsetting to a long-time fan of the comic books, like me. Though the stories have changed over the years, for the most part, Peter's character has pretty much been consistent. He is the quintessential "good man"; his personality is one we can all relate to at one point or another. He says in the second Raimi movie, "punch me, I bleed" and I think that perfectly sums up his character. I tend to care for a person like that and not a self-centered, angst-y, stuck-up, little sh17 that will bark at his surrogate parents that "they aren't his real parents" despite them raising him since he was born. Despite ASM throwing in some things from the comics that weren't in the Raimi movies, I amazingly still didn't care; none of the characters were developed and the story wasn't well enough for me to care.
I get that they're trying to appeal to a newer "generation" *cough*bankingoffofTwilight-kidswithneedlesslydarkovertonesthankstoBatman*cough* but I think there comes a point where you have to call it for what it is: Sony made this movie to keep the film rights to Spider-Man. Though there were ups and downs in the Raimi movies, at least for two movies, you saw that someone cared making them. These new ones I can't get behind because it feels like no-one cares. Keep the film rights, shovel some toys while you're out, and make money. Use 'em and lose 'em and wait for the next iteration to come out to peddle merchandise.