According to simple physics, running a mile or walking a mile should require the same amount of energy, and by implication that means it should make no difference to the number of calories you use whether you run or walk. But the problem with that argument, and it's one we've almost all subscribed to over the years, is that it doesn't really take into account the differences between the way the body works when you run and when you walk.
I haven't done a lot of reading around this at the moment, but I did come across some interesting research that has been done that suggests that we will have to concede that running does in fact burn more calories than walking. While simple physics might tell you that if you move a fixed mass over a fixed distance you will require the same amount of energy no matter how fast you move it, the physiology of our bodies means that we work (as in use energy) differently when we run compared to when we walk. We use more oxygen when we run and consuming oxygen requires energy which in turn raises our metabolic rate. One study found the overall difference in energy consumption to be around 100 Kj or about 25 Kcal (1 Kcal is what we would usually use as a dietary calorie).
So, if you can run, you will burn more calories over a given distance than you would walking it, but of course the point is that whether you run or walk, exercise is good for you. If by running you can only run say 3 days out of 7, but by walking you could walk 6 out of 7, then maybe, over the course of a week, you'd be better off walking.
The other thing that I want to look at, but I'm not sure I'll get the time or have the deep desire to do it, is the way some of the apps you can use measure the calorie burn of exercise. I use two apps, one for exercise, and one for food. The tow are linked and often report quite different results, probably because of the maths they use to make the calculations. That's okay, measuring my energy use is neither a high priority nor an exact science and certainly not the reason for using these particular apps. No, the algorithms aren't the issue, it's whether they take into consideration how much energy is being used above my basal metabolic rate.
So, yes we have probably got to concede that runners burn more calories than walkers over the same distance, but just because you're not a runner doesn't mean it's time to hang up your trainers. You might just have to go a bit further instead!!