What is the difference between walking and running?
Walking
Walking is a motion that can be simplified by the movement of an inverted pendulum. Walking has a time when both feet are on the ground, called double support. For the most part the motion is relatively passive, you just need your quadriceps muscles to keep you from falling to the ground and your calf muscle to propel you forwards with 1.2-1.5 times your body weight.
Running
Running on the other hand is better approximated as a spring mass system. Running has a time where both feet are actually off the ground called double float phase. Here it is all about energy storage and return. Your arch in your foot is one such spring mechanism, your Achilles another. Your calf muscle’s job is still to propel you forward, but now it must do so at 2-3 times your bodyweight. Please go home and give your calf some love tonight.
There is a whole other crowd of folks:
The Grounded Runner
These are the folks running 12 minutes per mile pace or slower, they have an inverted pendular motion that looks like walking and eek out a millisecond or so of a double support phase, but watch them and you will swear they look like they are running. Yep, they have us all fooled, look again, closely.
Treadmill vs Overground Running
Treadmill
No air resistance, so yep bust out the speed and fly. Want to mimic air resistance put your treadmill at a 2% incline and dang your workout just got harder. But remember when you are staying in one place, physics always wins and your braking impulse is going to be pretty darn close to your propulsive one. Yes, but it is so much fun to mess with your running mechanics when you can keep the speed constant!
Overground
GPS, Strava, Garmin go, watch the hills, the wind, the rain and make sure to look down once and awhile around the dog park. Country roads, trails, bike paths galore, just remember the camber roads all have to help with water drainage. Lots of variables to control, but that means you are always moving forwards.