What, if at all, is the difference of cooking an egg on high or low heat?

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.

Really just adding to @moscafj's answer with some personal observations:

If you use something like a duck or goose egg, you really see the differences due to the different protein structure. With them, at higher heat, the white will come out about the consistency of a rubber band. My experience is that a fresh chicken egg sees the same effect but less noticeable, and as the egg ages the protein the the white loses structure and the differences become more subtle. As a matter of taste, some prefer the firmer texture, others the softer, and some may not even notice. In my opinion though, the lack of difference between methods with he older egg is simply the result of lower initial quality.

In addition, frying at a lower temp I find can allow more heating and an almost custard like thickening of the yolk without cooking it hard. Again a matter of taste between an extra runny yolk that I would call not cooked at all, to warm but still runny, to cooked hard, and depending on which way you prefer your eggs you may not see a difference.

When heated, the protein molecules suspended in egg white unfold, straighten, bind with each other and precipitate out. The more heat and/or time is applied, the more this happens and the more the white solidifies. Meanwhile, the watery suspension medium evaporates.

See: https://chem.libretexts.org/Textbook_Maps/General_Chemistry_Textbook_Maps/Map%3A_Chemistry%3A_The_Central_Science_(Brown_et_al.)/13%3A_Properties_of_Solutions/13.6%3A_Colloids

I, like jpaugh, like crisp-edged (but not rubbery!) whites and a runny yolk, which requires quick cooking and high heat (I prefer olive oil for this). Crisping on lower heat would make the whites more rubbery and the yolk less runny.

On the other hand, Gordon Ramsay challenges Master Chef contestants to produce a consistent soft texture throughout the egg without browning or under- or overcooking, which is better achieved with lower heat and more time. By making it a speed/quantity contest, he encourages exactly the wrong approach.

Fast food eggs are rubbery because they are purposely overcooked for safety and because they often sit around under heat lamps. And they are often put in sandwiches, where solidity is desired.

It all depends on fat you're using. Small/low heat is good for butter. The egg will not be burnt on side, yolk will be creamy and runny.

Frying egg on high heat can be done with oil and there's a high chance it will have this brown outline, the white will be kinda chewy and yolk firm. It's the type of egg you get at fast food.

Difference in taste in noticeable with butter (and I mean butter and not god save the queen it's not butter) as it add sweetenes or saltines where oil fried will have, well, oil aftertaste.