I've been meaning for 50 years to address this. As players, the Stones might have had the edge, as they arranged for two guitars much more imaginatively than The Beatles did (or could, given Lennon’s being very limited), and Brian Jones, on both harp and slide guitar, was probably the most accomplished instrumentalist of all nine, but the Beatles were obviously light-years ahead vocally, and which Stones track can you say with a straight face rocks as authoritatively as “Money” or “Long Tall Sally”?
Frank Zappa, whose opinion some people valued, for reasons I was never able to divine, used to enjoy shocking interviewers by saying that he didn’t much care for The Beatles, and much preferred the Stones, by virtue of their being…blacker. So we were to understand that the Stones ever came close to as powerful a version of a black artist’s song as “Twist and Shout”? We’re to think that the Stones’ later reworking of The Temptations “Just My Imagination” deserves mention in the same breath as The Beatles’ “You Really Got a Hold on Me”? If Jones’s slide guitar was an indisputably potent musical weapon, Lennon’s voice was potenter by far.
Both groups were funny, the Stone unintentionally. Five little 104-pound twerps performing the music of the great-grandsons of slaves? Hilariously audacious! Mick Jagger, gangly in spite of his diminutiveness, aping the dancing of James Brown? Again, hilarious. I preferred the corny three-boys-on-two-mikes stage show of The Beatles, the homoeroticism of Paul and George shaking their pretty hair at each other.
As songwriters, there was of course no comparison, though the gap admittedly didn’t forever remain as wide as that between the puerile “As Tears Go By” and “Yesterday”. Keith Richards is said to have dreamed the famous riff of "Satisfaction," but what he really dreamed is the riff of Martha & The Vandellas' "Nowhere to Run." I'll take those in "Daytripper" or "I Feel Fine" any old time.
A lot of people were drawn to the Stones because of their greater air of danger and rebelliousness, as witness their not wearing uniforms. Keith Richards, who can be seen (decades later) on YouTube calmly unstrapping his guitar and trying to brain with it someone who joined his group on stage uninvited, was probably the most dangerous and rebellious of the nine, and of course we found out later that the nicely spoken, staunchly middle class (in the British sense) Mr. Jones thought it jolly good fun to beat up women, but Lennon had done no little wife-beating of his own, and probably wasn’t much less sociopathic than Keef (so referred to, FYI, because residents of the east end of London commonly pronounce th as f). I loved Marianne Faithfull’s guffawing incredulously when asked if Mick Jagger were really Lucifer reincarnate.
I periodically read on Facebook about what a fantastic guy Lennon was — kind of Gandhi, Martin Luther King Jr., and Nelson Mandela rolled into one, but a better singer than any of them. Well, I agree about the singing, and acknowledge that over the course of his tragically truncated life he might have become a very much kinder person, but there’s ample evidence that at least into his late 20s he was a horror story. He stole other musicians’ instruments. He mocked spastics and the physically deformed. (The most appalling cruelty I’ve ever witnessed was that inflicted on a spastic boy at my junior high school. Lennon, apparently, would have been one of his most avid tormentors.) After Beatlemania, his principal diversion was mercilessly belittling anyone who dared approach him at the London clubs where he held court. In retrospect, the messiah complex he developed with Yoko’s encouragement circa 1969 makes Bono’s look like a small, unambitious child’s. And he was an outrageous hypocrite, here railing bitterly about how he’d been mistreated as an escapee from the working class when in fact he wasn’t working class at all, there implicitly condemning material greed while renting a separate apartment in Manhattan’s Dakota apartment building in which to keep his and Yoko’s fur coat collection at just the right temperature.
Better than either? Best rock and roll band ever? The late-‘60s Who, hands down.