He doesn't want to be normal per se, but he wants to be loved and accepted, which is why he keeps trying so hard to prove himself as a superhero even if he occasionally backslides. At the same time, though, there's a big part of him that feels like he doesn't deserve to be loved and accepted, because of how guilty he feels for the lives he's taken in the past and his other less-than-noble deeds, which is why he can sometimes wind up sabotaging himself in much the same way Peter does (since like you said, Peter is a smart, talented good-looking guy who could have the world as his oyster if he really wanted it, so the only thing holding Peter back is Peter himself).
Peter's probably closer to having achieved the love and acceptance that Deadpool craves, which may be a huge factor in why Deadpool has taken to shadowing him as Spider-Man and trying to get him to be his friend, since if even someone like Peter can see Deadpool as a friend, it'll be a sign that someone like Deadpool deserves love and acceptance. It's pretty much the ultimate in validation, which is why, given my own sympathy to Wade and his issues, I really want to see him and Peter bury the hatchet. After all, Peter was the original outcast superhero of Marvel, which would make Deadpool kind of his nuttier, zanier kid brother as parented/influenced by DC's Deathstroke, and what kid brother hasn't wanted the approval and love of his big brother? (Yeah, I went there. Man the tomatoes!)