Pioneers like Jerry Lee Lewis, Little Richard and especially Chuck Berry were huge influences on both the Beatles and the Stones. Each used their influence in different ways. I would guess that most kids of the '60 first heard "Roll Over Beethoven" and the classic Berry rift as a Beatles song. Keith Richards emulated Chuck Berry. Form there as the '60s progressed, the Beatles and George Martin tended to lead the way as they moved from simple "Love Me Do" stuff to "Help!", "Rubber Soul", "Revolver" and into Pepper, White Album and finally self destruction, with "Let It Be" being the footnote and "Abbey Road" being the real 'out with a bang' finale.
You have to admit the Stone generally followed The Beatles, with albums like "Their Satanic Majesties Request" their attempt at a Pepper. IMHO, the Stones hung with The Beatles in the singles market, but the Beatles albums were more trend setting and trail blazing.
After 1970, the Stones continued on and maintained that singles domination while The Beatles split into 4 artists, all topping the charts at times, but by then the competition had really grown and in many cases surpassed both groups. Today, if you were not there in the '60s, it's easy to dismiss or ignore both of these groups as old acts who have over blown significance.
To answer the question at hand, I would have to say The Beatles were more influential because many more artists of the time (and in the following years) claim they were influenced by The Beatles than the Stones.
