Walking shoes are designed with the specific body mechanics and strike path of walking in mind. When you walk, your weight is more evenly distributed on the foot than compared to running; so your weight rolls from the heel, through the ball of the foot and continues to the toe from one foot to the other, over and over again until you stop and smell the flowers. This gentler, "rocking-chair-like motion" is accompanied only by the force of 1-2 times your body weight with each step, requiring your feet to absorb much less shock than the 2-3 times your body weight that running generates.
"Ohhhh... so it's a force thing!"
Essentially, yeah. Think of your runs. They don't call it "pounding the pavement" for nothing, right? Right. Walking shoes have greater arch support to protect where the force is heaviest on the foot. Running shoes, in contrast to walking, have more cushioning in the heel—the point of impact—and less protection through the ball of the foot.
The amount of heat generated during running is greater, too, so running shoes also are made with a higher amount of mesh to keep feet cool during exercise.
"So that's why so many running shoes have mesh!"
YES! But some walking shoes still like to include it. Your feet can get hot in any footwear, after all! The New Balance 847V2s below include synthetic leather and mesh in their upper offering durability and breathability.