Unhurried, the person can make sport having a better interaction with the medium perceiving more nuances. If it's pollution or another annoyance, it's better to go faster...
As it be, maths doesn't fail so the walking or running routine can end with similar health benefits and destining the same quantity of energy. Sometimes you see people running so slowly that others can pass them walking fast!
But perhaps the following elements are convincing enough to consider running instead of walking. First, anyone should be prepared for undesirable events where to run it's strictly necessary. Say a tsunami or a volcano spitting out lava may be an anecdote or a reason to lose everything. Walking usually doesn't work in this situation.
Once the organism is prepared to run, the time required for the exercise is less. Having too much for doing, walking is a bit stressing while running permits the benefit for health quicker, or simply to burn calories and responsibly control the weight.
Psychologically, it aids to improve the sense of effort and personal development in a solid manner. Those who run marathons know about this with no exception!
Finally any intense sport like running —in the sense of the word— during long distances really increases the performance of the organism, so activities that before seemed laborious and to sweat sure become different, so it's difficult feeling tired in everyday life. Instead you could feel the necessity of spending energy. The quality of life is very different when someone gets tired easily needing to rest or avoid an activity and others want to climb a mountain with the teeth like superman. Higher the overall strength and vitality there are more and better possibilities.