Typically the stance is that calories burned is a direct result of distance traveled. This is sort of true, but not really. For example, if you walk 10 miles, then you are walking a very straight line (one foot always on the ground) toward that 10 mile marker. When you run, you are actually jumping off the ground with every step and therefore you actually travel a little bit further when you run 10 miles than if you walk it. Since you traveled a little bit further, you burned more calories. Also, as you mentioned, running increases metabolism, and also, the body needs to repair more after running 10 miles then after walking it, so you are burning more calories there.
Not to mention that when you run, you hold your arms so that the point from the elbow to the hand is pretty much parallel to the ground. Casual walkers don't do that, and so are expending less energy just because of that. In order for a walker to compare even at all with a runner, they have to swing their arms while they walk the whole way, and keep a pace that is strong enough to make them sweat. Most people who defend walking don't do that. Go to any park and look at what you see - thinner runners and fatter walkers. Let's also not forget that you might be able to run 10 miles in about an hour, but it might take you 2 1/2 hour to walk it. At some point, running wins by time allotted alone.
Running wins. If that is your stance, then you win too.