Well, first of all, his identity as Billy Russo before the incident is important as it shows his roots in the criminal underworld. Then, the fight with the Punisher that led to his face being disfigured like a jigsaw is a central element. This not only changed his appearance but also his psyche. His new - born vendetta against the Punisher is also key. It's what fuels his actions and his character arc in the Marvel Comics. And his new name 'Jigsaw' which is a result of his disfigured face, is also an important part of his origin story.
In the Marvel Comics world, Jigsaw's story starts as Billy Russo, a mob - associated figure. His encounter with the Punisher was a turning point. The Punisher's violent act during their confrontation left Russo's face in a shattered state, looking like a jigsaw puzzle. This transformation was not just physical. Psychologically, it made him more vengeful and menacing. He became Jigsaw, a character known for his disfigured face and his hatred towards the Punisher. His origin story sets the tone for his actions and his role in the dark underworld of the Marvel Comics.
Yes, Jigsaw does die. His demise often comes as a culmination of his villainous deeds and the forces of justice prevailing. However, there might be alternate storylines or spin-offs where his fate is different.
It's not a straightforward answer. Sometimes Jigsaw dies in Punisher comics, but there are also instances where his fate is left ambiguous or he manages to survive for future storylines.