I usually see people frying eggs with butter and sometimes with soy oil - for example. But there are a lot of other ocasions in which these two options appear. My guess is that there are kinds of frying, for example: When you want to fry eggs, it's better to do it with butter, this is a kind of lighter frying. But when you are frying chips, it's better to do it with oil, this is a kind of heavy frying in which the food much receive more heat for a long time (I guess that oil can reach temperatures higher than butter).
1Related: cooking.stackexchange.com/questions/25469/…– Stephie ♦Dec 21, 2014 at 17:17
Basically it all depends on temperature:
Every fat or oil has a specific smoke point (see here for a list), at which temperature it starts to degrade. That means, not all oils are suitable for every heat, or all types of frying. Some types of oil should remain "cold", like extra-virgin olive oil.
Butter has a rather low smoke point (325-375 °F or 163-190 °C) due to milk solids. So, frying with butter is a bit tricky temperature-wise, but often butter is choosen for taste. (Clarified butter, on the other hand, has a smoke point of ca. 485 °F or 252 °C, which makes it usable for pan- or deep-frying. The taste is somewhat similar to butter, but not equal.)
As eggs are typically fried at a "low" temperature, using butter is fine. An example from the opposite end of the heat range would be asian cuisine, where food is prepared in a wok at very high heat. (I won't give recommendations what to use & when, as there are subtle differences, even between brands. If in doubt, read the label.)
So, to cut a long story short:
Decide what you want to do, how hot your pan is going to be and pick the matching oil.