The Beatles (or, at least, John and Paul) let the Stones record a cover of one of their own songs (I Wanna Be Your Man), producing the group's first big hit. In the frenzy of Beatlemania reaching the United States after the group arrived at JFK Int'l in February 1964, Mick Jagger himself admitted, at least, that the Beatles were far more popular than they were, and possibly even better than them. Later, in the famous 1967 broadcast of "Our World" featuring the first performance of All You Need Is Love, Mick Jagger can be seen in the background clapping and singing along alongside his girlfriend at the time, Marianne Faithfull (skip to about 2:35).

Though Mick Jagger cannot speak for the group as a whole, he himself was at least quite fond of the Beatles both as a group and as acquaintances.

Keith says this of the Beatles: "Everybody was talking about the Beatles versus the Stones and all that crap, and yet between us, it would be 'You come out first and we'll wait two weeks'. We would never try to clash; there was plenty of room for the both of us."

Although Charlie seemed to be a little more condescending: "The Beatles had a fantastic look, but they weren't very good on stage. They didn't do anything on stage—literally nothing: it was Paul going with his left hand and looking up and down and John nodding his head. Ringo moved more than anyone up there. The Beatles didn't swing live—but I suppose they didn't have to, they just sang 'Can't Buy Me Love'."

Source: According to the Rolling Stones edited by Dora Loewenstein and Philip Dodd