So Richard Parker genetically engineered spiders with human DNA, so that the venom could -for some reason- be used for a super soldier/healing serum. But the venom would only work on someone with Richard's DNA, because that's the DNA he put into the spiders.
In the first movie, they mention that Richard genetically engineered spiders, but I don't think they ever said anything about the purpose being for a serum. They also didn't mention that the spiders had human DNA. Oscorp were using the spiders to produce web, which is the web Spidey uses, so I assumed that the spiders were engineered for the purpose of producing web. In ASM2, they destroy all the spiders, which means that Spider-man would no longer have a supply of web.
What I don't get, is that if the spiders were engineered for the purpose of making this serum -which is implied in ASM2- then why would Oscorp have hundreds of the things in a web farming facility? What would make these spiders ideal for farming web if they have human DNA/magic venom? In the real world, we genetically engineer goats with spider genes to produce web, because farming spiders is impractical. This would have been a better idea for Oscorp rather than spending the money to engineer spiders that are still the same size as spiders, and thus wouldn't produce much web.
So yeah, it seems like they changed the purpose of the spiders. In movie 1 they exist for the purpose of producing web, which is kind of silly anyway, but in movie 2 the spiders exist for producing venom that can heal Norman Osborn. If they were made for this venom, then it doesn't make much sense to have them in the web farming facility, which is where Peter was bitten.