There are variations that take place throughout the bears, making one bear larger with white thick fur and one smaller with thinner and darker fur.
As they all compete for resources and survival, not all will have strong enough traits to end up on top.
The white bears thrive in their environment because of easy camouflage, thick fur and larger mass to keep warm, and bigger/stronger claws to catch prey.
As years of time and generations pass, the "survival of the fittest" comes into play. The animal that has the most beneficial traits for survival will thrive and the other will die off overtime. In this case, the brown bear has died off and the white bear will continue to successful live and reproduce.
The white bears have adaptations that are best fit for the environment.
These beneficial adaptations will be passed to their offspring and the offspring will be another generation of the successful species.