I never had much interest in either bands, because i started at a young age with bands influenced BY them. I heard the first New York Dolls album before I heard any Stones albums....so my way of thinking was "Why have the Stones when I could have an even rawer, sloppier, faster version of the Stones?".
I'd say another thing that happened was that Punk made me take more accessible things for granted. I knew that I could always go to a record store and find something by the stones or Beatles. I think my first Stones cd wasn't until I was like 19. I was working at a gas station, and some guy around midnight popped open his trunk and asked me if I wanted to buy some cd's because he needed money for the Newfoundland ferry.
In terms of the big 60's bands though, I always loved The Who
I like both the stones and beatles though, and i'm sure overtime I'll like them more and more.