Anytime I think of the Stones being more influential than the Beatles, I think of that one album they had that was basically their attempt at Sgt Pepper. It had a really similar cover and all the songs were psychedelic, but I can't remember the name of it. I don't know the circumstances surrounding that album, but it could probably be argued it was a sort of tribute to the Beatles, but in my mind that sounds really lame to just follow in their footsteps.
Also, The Beatles and the Beach Boys were always kindof in competition back and forth with the BB releasing Pet Sounds and the Beatles Sgt Pepper, with the Stones never really getting involved in that.
Another thing that comes to mind is how badly the Stones performed in their own film that they made, the Rock & Roll Circus, that they purposely blocked the release of the film for decades because they were totally shown up by The Who, who basically did one song and totally fucked everything up and did an amazing job.
http://www.youtube.com/watch?v=bCKlr5qeiQY
Admittedly, the only Stones album I've ever listened to in it's entirety is Sticky Fingers, which I didn't enjoy.