What is the difference between 기름 and 석유?
Do they both mean gasoline? If so, which is most frequently used?

Question

Answer

Nope. 
석유 is included in 기름. 기름 means oil in the broad sense(like all of greasy things)
And gasoline is a part of 석유

석유 = petroleum --> It means the whole things that were refined from crude oil.
원유 = Crude oil
(석유 includes 휘발유, 등유, 경유, ......and so on)
휘발유= gasoline
등유=Kerosene
경유 = Diesel Oil