It's only been ten years since Tobey Maguire and Kirsten Dunst starred in the first movie of their Spider-Man trilogy, and only five years since the last. I don't know anyone who was clamoring for another version of the same story, but if they were then this is probably not the one they were hoping for.
Peter Parker (played by Andrew Garfield) is pretty smart, but he frequently does very unintelligent things. When he develops a crush on Gwen Stacy (Emma Stone), it's understandable but unrealistic. When he tries to intervene in a fight between a jock and a fellow weakling, it's noble but painful. When he sneaks into the scientific research firm Oscorp while investigating the disappearance of his father, it's foolish and dangerous. But when he climbs into a strange machine full of lasers and spiders, he's out of his mind. Of course he gets bitten, and of course it gives him super-strength, super-stickiness, and super-douchiness. Fortunately for him the third one of these wears off, but not before it somehow makes him more attractive to Gwen. He doesn't get the ability to spin or shoot webs, but he just takes a machine to do that from Oscorp, makes himself a costume, and he's all set to fight crime.
His brand of vigilante justice makes him popular with the public, but the police don't share that opinion. That's especially true of the police captain, who just happens to be Gwen's dad. But when a giant lizard (who looks a lot like King Koopah from the awful Super Mario Brothers movie) starts to terrorize the city, the police force isn't particularly effective and it's up to Spider-Man to take him down before he completes his goal of turning everyone else in the city into monsters like him.
The Amazing Spider-Man really isn't so amazing. I do think that Andrew Garfield is a better actor than Tobey Maguire, and Emma Stone is better than Kirsten Dunst. I also like the mechanical web-slinger (which I believe is also consistent with at least some of the comics), but it's odd that no one seemed to connect the use of lightweight, ultra-strong material with the only company in the world capable of making it. It's also ludicrous that no one seems to suspect that the emergence of the lizard guy might be in some way connected to the research of the world's foremost expert on herpetology whose life's work has been trying to merge lizard and human DNA.
The new movie lacks the enjoyability of the previous series, and a big part of that is its unoriginality. It's true that both are kind of telling the same story, but it's up to the newcomer to distinguish itself from the existing work rather than just copying it. There are a lot of the same images between the films, like the need to save people dangling from a bridge, or the need for Spider-Man to get assistance from civilians. Despite having a different villain and a different love interest, nothing about the latest film seems fresh. If you're in the mood for a Spider-Man movie, then you'd be better off just watching the first two films from the last trilogy than going out to the new one.