I think they're apples v oranges (& didn't The Beatles always think of the Beach Boys as their real competition?). Comments credit the Stones with lasting power, which they deserve, but part of The Beatles's mystique was never growing old as a band, or releasing increasingly mediocre albums.
Reason they're apples v. oranges is the Stones are more traditionally structured: lead singer-frontman who also co-writes all songs. Whereas The Beatles were a true band: Each member sings lead on some songs & contributes compositions. You rarely saw them in interviews or on publicity tours not together, unless until near the end, e.g. John & Yoko in bed.
Beatles did embrace a wider variety of styles. Pop, sure, but also some fine R&R & blues-inspired songs.
I agree with those who see Paul's smackdown as helping out with publicity for the tour.