I can't recall ever seeing any differences on their menus. Their color schemes are the same. Both feature fresh baked burger buns. In both, workers hand make their burger patties. Both cook to order. To me, it doesn't matter that they seem the same. They both do a good job with the foods they prepare and serve.
I've seen only one Burnham Burgers. It's in Daegu. I've seen a Kraze Burger in Daegu, Itaewon, and at the airport in Incheon.