It was the Australian running coach Percy Cerutty (1895 - 1975) who posited that "You run on your legs, not with them".
The difference is easier to discern after close attention to how runners move.
Those who run with their legs have little reciprocal movement in their torso. All the locomotive effort appears to be happening from the pelvis down. It looks like hard work (because it is).
Those who run on their legs move very elegantly by comparison because their whole body is involved. The power is generated by the spine twisting the pelvis relative to the shoulders and that power in then transferred to the road by the legs.
As with so much in running a more subtle version holds true for walking. Whether trekking the hills or walking the tango, poised, efficient walking is a whole-body experience.
But if you literally could walk on your legs rather than with them, couldn't you walk with no legs at all?
As the gentleman in the video demonstrates, apparently so.