He was attending a Cubs playoff game by himself, and listening to the game on the radio, about as anonymous as you can be, then the whole world knows his name and face because of a natural reaction that everyone else around him also had. Everyone already knew Bucknor, and though it was more extreme after the error, he'd been in the Majors over 15 years at that point and had to have some experience with being harrassed by nobodies.
Bartman was just a nobody that became household name and hated in his hometown.
I'll take being in Bucknor's shoes every time.