We're going to go as lightly as possible. First we're going to start with running shoes having a total weight of about 1 pound 4 ounces instead of the 4 pound hiking boots. As Ray Jardine says, running shoes are to boots as sports cars are to dump trucks. The lighter the shoe or boot, the more maneuverability it provides. The more rugged the terrain with rocks and roots and steep slopes and difficult tread, the more maneuverability it requires.
Boots do protect the bottom of the feet when stepping on sharp-edged rocks.
With running shoes, though, it is much easier to avoid stepping on such objects altogether. When avoidance is not possible, the shoes allow us to tread more gently. And incidentally, running shoes impact the earth far less than stiff-soled boots. In addition, the biomechanics of walking comes in to play. The heavier the footwear, the more energy required to draw the foot ahead with each step.