Spider-Man got his powers after being bitten by a radioactive spider. This gave him enhanced strength, agility, the ability to climb walls, and a 'spider sense'.
His sense of humor might change. While he still has his quippy nature, it could be more cosmic - themed. For example, he might make jokes about black holes or alien cultures. His empathy could also expand as he meets different beings from across the universe and understands their plights.
If Spider - Man kept the cosmic powers in fanfiction, he would likely face new challenges. With great power comes great responsibility, and these cosmic powers might attract more dangerous enemies. He could also struggle with controlling the powers at first. His relationships with his friends and family might change as they try to adjust to this new, super - powered version of him.
In some fanfictions, Spider - Man might gain the ability to teleport across the cosmos. This could lead to him exploring different galaxies and encountering alien species that are both friendly and hostile. He could use his spider - sense in new ways, like detecting threats from across the universe.
His character could become more confident. Having cosmic powers would give him an edge in battles and in general life. He might start to take on more global or even universal threats, expanding his role from just protecting New York City. His self - identity might also shift a bit as he is no longer just a normal guy with spider - like abilities but someone with cosmic - level powers.
He might first go into denial. He could keep trying to do the things he used to do with his powers, like swing from building to building, only to fall. Then he could start to rely on his intelligence more. Since Peter Parker is a smart guy, he'd use his scientific knowledge to try to find a way to get his powers back. He may also look for allies who can help him, like other superheroes or scientists.
Deadpool got his powers through a messed-up experiment. His body was subjected to a procedure that gave him superhuman abilities, but also left him scarred and crazy.