Both bands started out as rock 'n roll bands. The Beatles evolved into an art-rock band (for lack of a better term).
The Stones did not.
In my opinion.
I can claim quite a collection of albums by both The Beatles and The Rolling Stones, and I enjoy both, still, after all these years, though I prefer The Beatles.
I've long thought of The Beatles in terms of "song crafting" and the Stones in terms of "performance". Early Beatles and early Stones have a similar sonic texture. The Beatles got past this when they started recording albums of their own material; they seem to have spent creative energy to craft each song as a unique gem. The "song" becomes "the thing", and it is more than just the progression of notes, the tune, and the lyrics; it is also a matter of the orchestration. The Stones are rarely subtle in their orchestration; it remains generally unchanged from song to song, so that their sound is strongly identifiable. One thinks immediately upon hearing the Stones: "That's the Rolling Stones playing". One thinks upon hearing the Beatles: "That's a Beatle's song."
Many cover Beatles' songs, yet those covers always seem somewhat lame compared to the original, because each Beatles song is carefully designed, as a fine object of art is, and any change of any part makes for a marring of the original. One can cover a Stones song, and the object is to get the
sound of the Stones, not so much the sound of the song, since there isn't one.
I suspect that if the Stones recorded an album of Beatles songs, it would sound like a Stones album, with the textures of every song having a homogeneity. I suspect that if the Beatles recorded an album of Stones songs, we would finally have opportunity to hear what each song has to offer individually, in its own universe of sound.
By my measure, I can likely conclude that the Stones is the stronger "band". But I prefer hearing songs to bands, and thus follow first the path of the Beatles. Again, I turn to the Beatles when I want to hear Beatles songs; I turn to the Stones when I want to hear the band play. It's been that way for me for years; I don't expect much to change.
On my Discogs database, the term "Beatles" gives me 74 hits, which includes several complete album box sets and many rarities.
"Rolling Stones" provides for 17 hits, but includes two large multi vinyl disc collections of their albums.
In my collection.