^Well, first off, we've established that Bruce is not a "madman" or a "lunatic." He doesn't meet any of the diagnostic or legal criteria for insanity, and his adoption of a costumed crimefighting career is actually fairly normative behavior in the context of the DC Universe. He is troubled, obsessive, driven, single-minded, fixated, but not insane.
Second, Peter was fifteen when he lost his uncle, and he had the loving support of Aunt May. Bruce was seven to ten years old (depending on the version) when he had both his parents ripped from him, and though in some versions he was raised by Leslie Thompkins and in others by Alfred, it still wouldn't have been the same. (Yes, Peter lost both his birth parents, but it happened when he was too young to remember them.)
Third, unlike Peter, Bruce directly witnessed the murders. They happened right in front of him. That's bound to be more traumatic.
Fourth, Bruce was a pampered rich child who was exposed to the concept of loss very suddenly and drastically. Peter was a bullied, unpopular student and his family never had much, so he was more accustomed to dealing with setbacks in life and using humor to cushion his psyche. So he just had better defense mechanisms when the tragedy happened. Bruce had to build a whole set of defense mechanisms from scratch after the tragedy, and those defenses were shaped by the tragedy and by the dark, dangerous world it forced him to discover. As such, his coping mechanisms are somewhat more drastic.
Fifth is environment. Peter found himself facing a New York that wasn't too different from the real one, maybe a little cleaner and nicer in some ways, and that already had its share of costumed adventurers to help keep it safe. As Bruce grew up, he saw Gotham becoming increasingly plagued with corruption, violence, suffering, and despair, and the authorities who were supposed to fix things were an integral part of the problem. So he would've grown up with more of a siege mentality.
Sixth, maybe they were just different personalities intrinsically. Some people are predisposed to depression and a more cynical worldview, others predisposed toward a more optimistic approach. Bruce probably would've grown up to be intensely focused, serious, and driven about his undertakings even if he hadn't suffered a tragedy.