Bardeaux
Bluelight Crew
The Beatles were clean cut, nancy boys whereas the Rolling Stones, they were down right dirty!
Depending how you like your druggy sway, the Rolling Stones were all all hard core...man 8) oops up yeh :D
You do realize you're talking about a band that traveled to India, slept in tents while taking copious amounts of LSD, grew their beards for months and refused to get out of bed (yet still managed to record music) for just as long, yes?
But I digress, we already covered shtick vs. the actual music

It feels like I'm just wanking it now, but I also forgot to mention the subject matter found within the music of The Beatles. The early days were cutsie by today's (even by mid 60's) standards, sure. But the attitude changed because the beatles changed. As mentioned above, it's all about context. In 1963, Lennon told a crowd of thousands (including the Queen) "If you're sitting in the cheap seats, we'd like you to clap our hands a long to this number. If you're up on the balcony, just rattle your jewelry!") Even during this early period, before She Loves You, before Ed Sullivan and I want to Hold your Hand, the Beatles were high on speed and openly acknowledging class warfare in England.
When the Stones were releasing Aftermath, The Beatles were releasing Revolver. So many new innovations were being discovered (such as the backwards guitar that Hendrix would make famous), so many new ideas were being introduced to a mass western audience (Tomorrow Never Knows being heavily influenced by the Tibetan Book of the Dead), John and George's Eastern philosophies, string arrangements like on Eleanor Rigby weren't even within the grasp of the stones at that time.
Gimme Shelter is indeed one of the greatest tracks of all time. But compare it to the Abbey Road Medley. The complexity, the emotion. The Beatles could be fucking raw. Yer Blues, I believe, anyway, was a jab at Mick Jagger & Co. The composition of While My Guitar Gently Weeps, and Clapton's solos over it was just unmatched at the time.
If it's rawness and intensity you're after, look into Lennon's solo work. Cold Turkey, Well, Well, Well, I found out. This was extremely harsh material. And had the Beatles gone on into the 70s, some of that radical (politically and musically) oomph may have crept into the Beatle's discography. It's important to remember, the Beatles were pretty much dunzo by the time Brian Jones died.
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