Why do kids run, while adults walk? Is it simply a matter of the fact that adults’ walking speed is generally greater than kids’, so kids feel the need to run to make up the difference? Is it a matter of youthful exuberance, energy, or immaturity? Is there a factual answer to this question?
Children lack patience
…and possess invulnerability to falls.
Physics. Children have less mass and shorter legs, which helps them produce short bursts or speed. Adults are faster, but it takes them longer to accelerate. It’s why a man can outrun a horse for the first 10 yards or so.
I can remember running to school and thinking, “walking’s for losers. It takes too long.”
because they’re kids? 
Let’s look at this the other way … why do adults walk where children run? … because adults are lazy? … or just simply kids are excited to get where they’re going and adults have already been there before …
In a group of people walking somewhere, the adults usually set the pace.
An adult walking pace is often in-between a kid’s natural walking pace, and their natural running pace. So the kids get into a cycle … plod, plod, plod, fall behind … ZOOM to catch up…plod plod plod, fall behind…
That’s what happens in our family, anyway. Not with everyone - when I used to look after the kid next door after kinder, I was blown away by how much of a natural zoomer she was - about twice as fast as the Small Boy. Same age. Go figure. Her mum was a whirlwind of energy too. It’s pretty harsh when you’re struggling to keep up with a four-year-old.
Why are we running? I don’t know but there could be ice cream!!!
[Eddie Izzard’s line, not mine]
I think toddlers find it easier to stay balanced when trotting than when walking. No cite, though.
I’ve always presumed that it has something to do with metabolism. Kids just seem to have so much energy, and if they can’t burn it off, they get all sorts of antsy.
Same answer in different words: It’s for the same reason that adults love to just relax, but kids always need to be moving.
Not a complete answer.
One area you might expand on is the ratio of mass to linear size…
Basically adults are twice their height but 4 times their weight… its more to hold up and more effort. The kid can barely carry a 10 kg load ,but they can easily move their 30kg around at child speed.
If you just learned to walk, then run, wouldn’t you show off, too?
In a kid’s mind, someone or something’s always chasing you.
Especially a giant SHAAAAAAAAAAARK…! on dry land.
Adults forget how invisibly stealthy they are, and stupidly walk around not knowing how close they are to being eaten at any moment.
Personal theory: Kids and adults each have the same amount of energy, but in kids it’s in a smaller package and causes them to respond and move more energetically to release it.
Irrational exuberance. 
And remember, Science tells us that moving quickly is correlated with greater life expectancy.
Slowpokes beware.
“Youth is wasted on the young.” - G. B. Shaw (sort of)
Because they are allowed to?
Seriously, if jogging or running were not frowned upon in general settings, I would be right with them. Walking from place to place just takes so long–it is wasted time.
Another approach: let’s assume that the leg behaves like a physical pendulum when you are walking with least energy waste. You can show that the period of the pendulum will scale with length^2 (assuming equal shape and density). So a child with legs half the length of yours would tend to take four steps for each of yours.
I think it’s more due to brainpower than muscle power.
Adults have functioning brains–which plan ahead and tell their bodies to conserve energy because they have to stay alert for the next 14 hours.
Kids have kiddy brains…which run at full speed all the time, with no worry about planning ahead. Then,when they run out of energy, they just stop wherever they are, and recharge-- by falling asleep on the nearest adult shoulder or lap.