Believe it or not, in a perfect system (i.e. exercising in a vacuum) you would burn the same amount of calories if you ran a mile or walked a mile. Think back to your high school physics days when you learned the simple equations that supposedly governed our universe. For example, Work = Force x Distance. The number of calories burned depends primarily on your weight and the distance traveled. This means measuring the energy required to move a particular weight (you) a particular distance.
Of course running a mile is going to be more beneficial for your overall health than walking in other aspects. While jogging or running, you are positively stressing your cardiovascular system, bolstering its efficiency at delivering oxygen and nutrients to your working muscles. Running also elevates your metabolic rate dramatically. This raise in metabolic activity will last for hours and your body will continue to burn calories during this time until you eventually come back down to a basal level. Keep in mind though that if you can’t run because of knee or joint problems, you can still get just as lean by walking!!