Something interesting to me because I grew up through a lot of it, and which I understand people may not care about, is how ingrained they are with the culture and zeitgeist of the time period. The Crossfire Hurricane doc goes into this in some detail; the Beatles were the white hats. The Stones, looking for their niche (and despite being pretty good friends with the Beatles, especially John) sort of defaulted to what Keith called "the grey hats". But over time, as they moved into the late 60's, early 70's, and the Beatles were not so clearly the white hats anymore, teh Stones sort of started to, as Mick said, live as their characters, not as themselves. That led to the drugs, Altamont, and was reflected in the music. Someone said "antiwar, we get it" about Gimme Shelter, and it was so much more than that.
Then go into '71, '72, and becoming literal exiles to their country (they famously recorded most of Exile in the south of France, until they had to leave because the law was on to them), it became something so much more than the music. THEN they took that debauchery and hedonism on the road... that Exile tour is one of the most famous in rock and roll history, and documented in "Cocksucker Blues" a documentary so incendiary that the band fought in court to block it's release (it's never been widely released, though it HAS to be shown - by court order - a select number of times each year under strict circumstances, i.e. like at art festivals or in museums).
People often talk about how the Beatles were "John's band" for the first four years, and "Paul's band" for the last four. The Stones take that to another level. The band morphed incredibly over the 50 years or so they've been around. Initially Brian Jones' band, by the mid-to-late 60's, Jagger and Richard had sort of marginalized Jones, and it was really Keith's band up through Exile. Exile - recorded in his basement in France - was his last real burst of creativity before succumbing to a heroin addiction. Jagger "led" the Stones through much of the 70's, until Keith kicked his habit in '78 or so. The 80's albums - Emotional Rescue through Dirty Work - were the band fighting (the last two, Undercover and Dirty Work, the two rarely spoke and didn't tour for DW because of the acrimony). And from Steel Wheels to the present, it's been a consortium. The interesting thing from the Crossfire Hurricane doc and the Live Licks docs is how involved Charlie is; he picks a lot of the artwork, and was instrumental in designing the stage sets for the tours in the 90's and 00's.