If Mr. White had listened to the sergeant, how might the outcome have been different?

In addressing this kind of question, it's important to note what kind of advice the Sergeant-Major provides. He gives a warning and gives details concerning the paw's history. He advises White against making wishes on the paw and asserts that it is better off destroyed.
The simple answer to this question is that, had White followed this advice, he should have at the very least resisted the temptation to make that fateful wish. If he had not made that first wish, his son would have survived and the events of the story would have been completely averted.
"The Monkey's Paw" ultimately exists as a work of fiction. With that in mind, providing an answer to this kind of question can easily amount to a creative exercise. After all, one can easily imagine any number of potential stories which all proceed from a similar starting point, in which Mr. White either determines not to make a wish upon the monkey's paw or perhaps even decides to destroy it entirely . . .