Remember the scene in Amazing Spider-Man where the Lizard says to Peter Parker, "You want to know the truth about your parents? Come and get it"? Us neither. That bit was in a bunch of the trailers for the film — but got cut out before the movie hit theaters, joining many other instances of scenes that are in trailers but not the actual movies.
Actually, the trailers hinted that there would be a whole storyline in the film, in which Peter Parker discovers a deeper, darker truth about his parents — one that affects him personally. I was kind of leery about this notion, especially with the hints that Ma and Pa Parker genetically tweaked Peter to give him spider DNA — and that's why the spider-bite affected him like that. It seemed sort of similar to what Ang Lee's Hulk did, making Bruce Banner's dad's experiments on Young Bruce the cause of his Hulkitude rather than just the Gamma radiation.
And meanwhile, you might have noticed that some plot threads get dropped rather unceremoniously in the final cut of Amazing Spider-Man — there's the SWAT Team that gets turned into lizard people, which seems to be leading up to a big set piece that doesn't materialize. (They even made a playset for it.) There's the fact that Irrfan Khan's character, Rajit Ratha, is on his way to the Veterans Administration hospital to test Curt Connors' cross-species genetics juice on some unsuspecting patients, when Connors attacks the bridge. And then Dr. Ratha is never seen again.
(There's also the fact that Spidey stops looking for the guy who killed Uncle Ben, but I actually feel strongly that that's not a dropped plotline — Spidey grows up and starts helping people instead of looking for revenge. It's spelled out pretty explicitly.)
In any case, over at Badass Digest, Devin Faraci does a great job of analyzing some of the scenes from the trailers that aren't in the movie, to figure out what sequences were cut at the last minute. I'm not going to summarize it here — you should just go read it for yourself. In a nutshell, there was a whole sequence where Dr. Ratha finds Connors' sewer lab and gets killed. But more importantly, there's tons of evidence — including snippets still in the final movie — that the film included a whole "Peter's parents borked his DNA" subplot that got taken out after the studio saw a rough cut.
And really, that's a very good thing. As I mentioned above, I was super apprehensive about the hints in the trailers that we were going to get a radical reinvention of Spidey's origin story — one in which he was destined from birth to become Spider-Man, or at least the only person who could have become Spidey. As it is, the fact that Peter's parents were directly connected to the experiments that led to his spider-bite seems too convenient, and makes his world too small. (Although on the plus side, Peter's not bitten by accident — he actively goes looking for answers, and gets bitten as a result of his decisions.)
Spelling out that Peter Parker was genetically engineered to be Spider-Man — or to be capable of becoming Spider-Man, at least — takes away some of the "normal guy gets superpowers" angle. It also turns his parents into actual monsters, who experimented on their own baby, and possibly sets up a scenario similar to Ang Lee's Hulk where his parents turn up alive, and sic spider-poodles on him. (It seems pretty clear that in Marc Webb's Spider-verse, the Parkers are alive.) I kind of hate the idea of making Spidey's origin so overdetermined.
But also, you know, there's just the fact that Spider-Man's origin, like a lot of the best Stan Lee creations, is incredibly simple — except in its ramifications. Spidey's single spider-bite (plus his miraculous invention of web-shooters and so on) gives him the power, and then the interesting question is what he does with that power. Making the mechanics of Spidey's power-up more complicated just messes with what makes him such a classic hero — so it's a very, very good thing that Sony slashed and burned that subplot from Amazing Spider-Man. Let's hope it's kept out of the sequels, too.