Jasper being that insane, instead of more like Marcus, Irina, or Bella, is a direct consequence of his personality, the nature of his and Alice's relationship, and the exact circumstances surrounding his particular situation, including particularly the experience he refers to when he says he felt Alice die.

Everyone reacts to their mate's death differently, and how said mate dies matters a lot. (For instance, Maggie would have had a unique reaction if she'd eaten Gianna when they met, versus if Gianna had died trying to carry Elspeth to term. And if Maggie had finished crushing Gianna when Gianna was turning, that'd be a whole nother can of worms yet.)