The proteins in egg white and egg yolk behave differently at different temperatures. It is an ingredient that responds to very subtle temperature variations. That is why it is a favorite item to cook for those of us interested in low-temperature cooking using an immersion circulator (sous vide).
However, predating the immersion circulator, the Japanese lowered eggs into the Onsen hot springs, cooking them low and slow, to produce a texture that was not able to be produced any other way. More recently, Dave Arnold created a handy chart that illustrates eggs cooked sous vide at various temperatures.
Of course, when cooking in a pan, you have less precise control, however, you can certainly come up with different results by using high heat, medium heat, or low heat. In fact, ChefSteps has instructions for a "fried" egg that they called the "emoji egg." It uses very low heat and takes several minutes.
All of this to say is that you are able to control the texture of the white and yolk with subtle variations in temperature. Also keep in mind that at higher temperatures, the browning of fat in pan and egg white will contribute to flavor. There is also an added bonus of lower temperature egg cookery for some people who experience, and are turned off by, the sulfur aroma that eggs cooked with higher heat have. It turns out that cooking below 72.5 C (162.5 F) keeps these aromas at bay.