Here's the way I look at it: They didn't really need to tell the origin story again (another Spidey-tale, with a new actor, would've flew fine), but they did a pretty good job of doing it. Better than Raimi and his team, in some cases. Not all, but some.
The rest of the cast are all quite good, Ryhs Ifans is striking and very good as Dr. Curt Connors/The Lizard. Hitting the right notes of a good man who gets himself in over his head. The only sad part here is that Dylan Baker never got to fully flesh out the character he played in two of Raimi's films.
I really, really loved Sally field and Martin Sheen as Aunt May and Uncle Ben. Something about how they were handled in this film worked so much more for me than what Cliff Robertson and Rosemary Harris did in Raimi's film. They're all wonderful actors, but what Webb and team pull off with these characters worked much more on me, emotionally.
The Amazing Spider-Man is not a bad movie, and anybody who tried to tell you that is overstating things. It works. Is it too soon after Raimi? Maybe. I do encourage you to check it out for yourself.
At the end of the day, for myself, personally, I have to agree that there was no need to re-tell the origin story. There is an element where you're just waiting for the events to fall into place, again. Still, as with every time a comic-book is written re-telling that story (which happens a LOT), the way it's told trumps everything else. This is a well-told version of Spidey's origin. It's exciting, funny, well acted and directed. It's ludicrous to think that serialized characters like this are not going to be told, and re-told, and re-told again. It's the nature of the characters. Every generation gets their own version.
That said, guys? Next time, just skip re-telling the origin.