The mutation disease affected Spider-Man and caused him to transform into a giant spider creature which the media dubbed Man-Spider.
The first symptoms of the disease caused Spider-Man to lose his spider powers. To figure out what was going on, Spider-Man went to see Dr. Curt Connors. After Connors ran some tests he discovered that his DNA mus mutating even further into something that wouldn't be human. However, Connors was unsure what he was mutating into.
Spider-Man heard that Professor X was the world's leading expert on genetic mutations and went to see him in hopes that Xavier could cure him. However, Professor X told Spider-Man that his job was not to cure mutants but to help them control their powers an accept who and what they are.
Dr. Mariah Crawford later created a serum that she believed would cure Spider-Man of his mutation disease. However, Dr. Crawford was unsure if the serum would cure Spider-Man or accelerate the mutation. Against Dr. Crawford's warnings Spider-Man took the serum. However, the serum ended up accelerating the mutation and Spider-Man grew four extra arms.
Not long after Spider-Man fully transformed into a giant spider monster the media dubbed Man-Spider. Man-Spider was then hunted by Punisher and Kraven the Hunter. Punisher wanted to kill Man-Spider while Kraven wanted to help Mariah Crawford cure him. In the end Punisher and Kraven captured Man-Spider and Mariah Crawford was able to give Spider-Man a serum which transformed him back into a human.
However, Spider-Man learned that the serum only reversed his mutation but did not cure him.
After his first transformation, Spider-Man would often feel pain in his side and would need a dose of radiation from the Neogenic Recombinator every twenty four hours to prevent from turning into the Man-Spider again.
When the Tablet of Time was discovered, Dr. Connors believed that it had the power to cure him. However, they never got the chance to use the Tablet of Time because it was stolen by Silvermane and was believed to have been destroyed in an explosion.
However, the Tablet of Time survived and Hammerhead sold it to Adrian Toomes. Toomes studied the Tablet and created gauntlets that allowed him to drain a persons youth and make himself younger. When Adrian Toomes, now called Vulture, drained Spider-Man's youth he inadvertently absorbed Spider-Man's mutation disease into his own body. Curt Connors promised to cure Vulture. However, he lied. Connors instead returned Spider-Man's youth while keeping the mutation disease in Vulture's body. This event cured Spider-Man of the mutation disease.
Spider-Man was once secretly followed by the Green Goblin. Unaware that the Green Goblin was following him, Spider-Man's spider sense went crazy. Because of this Spider-Man feared that his mutation disease was returning.
In the comics
In the comics, Peter drank a serum that was suppose to remove his spider powers. However, the serum had a reverse affect and caused him to grow four extra arms. This was changed in Spider-Man: The Animated Series to where the serum was meant to cure his mutation disease but instead accelerated it.
Another difference between the comics and Spider-Man: The Animated Series is that in the comics Peter only grew four extra arms while in Spider-Man: The Animated Series, Peter Parker became a giant spider creature.