When Peter Parker was bitten by a lethally irradiated spider, radioactive mutagenic enzymes in the spider's venom quickly caused numerous body-wide changes. Immediately after the bite, he was granted his original powers: primarily superhuman strength, reflexes, and balance; the ability to cling tenaciously to most surfaces; and a subconscious precognitive sense of danger, which he called a "spider-sense."
Several biologists on the History Channel’s Spider-Man Tech stated the effect of a radioactive spider bite (if any) would not be nearly enough to cause a mutation in a human body. However, they said the use of "genetically engineered" spiders from the live-action movie and the Ultimate continuity was more plausible. Instead of radioactive venom, the bite would have to carry a powerful retro virus (similar to HIV) that would spread through the body by taking over neighboring cells and actually becoming ingrained in the person’s DNA.
This meaning that the Spider-Man we know could not really exist.