Robert, the Stones started out as an R&B band, the Beatles started out as a R&R band. I always liked them both, and, except for the "British Invasion" thing, I never looked at them as "competing". Two different genre' of music from the same global neighborhood.
I saw the Beatles at Shea Stadium once. Couldn't hear them well because of the screaming teens, and saw the Stones play about half a dozen times.
The Stones always had great warm up acts, too. Last time at the Garden the Opening act was Albert King, the second act was B.B. King, and then the Stones (who played for nearly 3 hours). The point is each and every Stones Concert had, at least, one blues artist in the warm up position. It harkened back to their roots.
The Beatles were more refined and polished, and I liked that, too. The stones were more "guttural" in their music and in their performance. There was even a difference in their stage appearance. The Beatles, when they were still touring and performing dressed alike in suits, while the stones essentially introduced American to the street clothed, garage band look.
I really like both bands for what they were, and the two very different musical styles they had (have).
