Why do some recipes require or insist on the use of a wok over the use of a skillet / pan?
There are a couple of reasons, traditional and some functional:


The home cultures where these recipes are indigenous use a wok, so many recipe authors go the same way
Woks are usually made out of carbon steel, and are poor conductors of heat.  This means that the strongest heat from the concentrated heat source is in the center/bottom of the wok.  As you go up the sides, the level of heat decreases.

This permits staging ingredients up the sides of the wok to stay warm or cook slowly, while the ingredients at the center of the wok are being cooked intensely.


In practice, the real issue with woks is that they are properly used over a very intense heat source.  This permits the stir fry technique, and its unique wok hei flavor to develop in the food.   Note that it is this intense heat (and the carbonization of the seasoning layer on the wok) that actually create wok hei, not the shape of the wok itself. 

Western home cooking equipment usually cannot get hot enough to do this, whether you use a wok or a flat bottomed skillet.  Still, over a standard electric burner or even a home style gas burner, using a flat bottom skillet may in fact let you get better heat transfer into the food, and come closer to genuine stir frying than you could with a round-bottomed wok.

Even for dishes that suggest you push food up the sides of the walk, you can use a skillet if you choose:  you cook the food in stages, removing it from the pan instead of pushing it up the sides. Combine the ingredients the end to marry and finish cooking, much as when they are tossed together in wok-based cooking.