My thesis: The Stones at their best are better than The Beatles at their best.
Top 10 Rolling Stones Songs:
Dead Flowers
Rocks Off
Street Fighting Man
Sway
Ruby Tuesday
Wild Horses
Brown Sugar
No Expectations
She's A Rainbow
Let It Bleed
Top 10 Beatles Songs:
If I Fell
Rain
And Your Bird Can Sing
Dig A Pony
I'll Follow the Sun
Strawberry Fields Forever
All My Loving
In My Life
She Said She Said
Two Of Us
I'm struggling to find a solid case to put any of the Beatles songs in the Top 5 over any of those first 5 Stones songs.
The Rolling Stones had 3 or 4 good-great albums depending on who you ask, and they were all within a 4 year period: 1968-1972. Other than that, they have a handful of really good songs before 1968 and after 1972. The Beatles had probably 8 or so good-great albums depending on who you ask, and they were all within a 7ish year period.
In terms of longevity, The Stones have been around waaaaaaay longer, but being around longer does not mean that they are a better band especially considering the RS peaked in the early 70s and most of what they've released since then has been mediocre other than a few standout songs (Beast of Burden, Start Me Up).
My personal conclusion: The Beatles' body of work is far more consistent, and it's not even a contest. They wrote lots of good-great and clean sounding pop songs, successfully tried out different styles, and stayed relevant. The Rolling Stones have an inconsistent body of work. For the most part, they were at their best when they stuck to rough around the edges heartland rock and roll. The Beatles wrote a lot of good songs, a few great ones, and very few bad ones. The Rolling Stones wrote a lot of mediocre songs, but had a string of great ones. The highs of the Rolling Stones were higher than the highs of the Beatles. The Lows of the Rolling Stones were lower than the lows of The Beatles. The Beatles operated on a 7 or 8 out of 10 throughout most of their career and only once in a while would reach a 9 or a 10. The Rolling Stones on the other hand operated on a 5 or a 6 throughout most of their career but were able to reach the 9s and 10s more often than The Beatles. The Beatles had many more good songs, but the good songs that The Rolling Stones had were better than those that the Beatles had.
If you put me on a desert island and tell me I can pick a 10 song playlist of only Beatles/Stones music, I'd probably pick 6 or 7 Stones songs to 3 or 4 Beatles songs. The Rolling Stones tend to ignite something in me that the Beatles don't as often. They are both great bands worthy of our respect and admiration for different reasons, but I'm going with The Stones.