I'm not really bothered by the radioactive/genetically mutated spider thing.
Nuclear science was the cutting-edge, mystery science back in the 50's. Genetic engineering is like that today. And (according to comic book science) what would radiation do but alter your genes to give you super-powers? We all know radiation doesn't do that (except to cause cancer), so no biggie to make the spider genectically-altered.
When this movie was being filmed, the superfans went apeshit over the webs being spun from glands in Parker's arms, instead of artifical webslingers he built himself.
In the end, it really doesn't matter, does it?