The running would burn slightly more calories for 2 reasons. Running is less efficient than walking and your increased heart rate would last a little longer after you finished.
For me I just walk and don't worry about the extra few calories I burn since I figure I MIGHT burn some muscle too if I run.
By the definition of a calorie running or walking would be pretty close.
A calorie is the amount of energy needed to raise one gram of water one degree centigrade.
And the definition of work is something like the amount of energy needed to move a set amount of weight a certain distance. So since you are moving the same amount of weight(your body) the same distance it is APPROXIMATELY the same.
Kind of like the riddle which is heavier a ton of iron or a ton of feathers
walk 15degree incline at 3.5-4mph will burn more fat do a gas exchange test-it will give you an exact heart rate to be at to burn maximum amount of fat.