The Old-Timer is an experienced man who has survived the Yukon winter nights, and his advice is sound: don't travel alone, especially in severe cold. However, the protagonist feels himself superior to the advice of experience, since he is an individualist.
That man from Sulphur Creek had spoken the truth when telling how cold it sometimes got in the country. And he had laughed at him at the time! That showed one must not be too sure of things.
(London, "To Build a Fire," eNotes eText)
Their main difference is that the protagonist trusts in himself alone to survive the long walk, while the Old-Timer has learned that the bitter cold is more powerful than any one man. The Old-Timer's knowledge of the cold supercedes his ego, while the protagonist's ego overpowers his reason, leading him to attempt the walk alone and without proper mental preparation.