To be honest I have never really understood a question like this because it can't really be answered. Music is such a personal matter of taste there really is no right or wrong answer. I hope that doesnt sound rude it's not meant to be.
Personally The Beatles and Pink Floyd own me. Depending on my mood depends on who I listen to at any given time.
As for The Beatles or The Rolling Stones, I seriously like the Stones but The Beatles? I could listen to John Lennon sing forever and never tire of it. I pick out every drum beat, bass sound, rhythm and lead guitar sound and appreciate each members contribution on an individual basis.
Like someone else said "I enjoy Beatles music, but I know the Stones" same for me but the other way around. I adore The Beatles