Polar bears climate is shrinking, while grizzly climate is pushing north thus expanding, so due to habitat the Grizzly is better adapted to come out on top.
If they meet and have a fight, most times the polar bears comes out on top, it is bigger and stronger. This happened when the flat faced bear met brown bears, but with time the flat faced bears grasslands disappeared so it died out, while grizzly habitat expanded. The same thing is occurring with one major difference; brown bears and polar bears are very closely related, thus they are mixing.
When brown bears tried to cross to north America from Eurasia in ancient times, they always ran into the road block of flat faced bears, but some brown bears were able to adapt to the arctic, these were the first polar bears. Flat faced bears did not live in the north pole.