Were the beatles and the rolling stones rival bands?

did they not like eachother? i remember reading the beatles actually giving a song as favor called I Wanna Be Your Man to the stones. but i heard in 1971, Mick Jagger had made some diminishing and insulting remarks about the beatles. and John Lennon stroke back saying (and i quote) "Jagger and his band weren't in the same class, musicwise or powerwise, never were" did they hate eachother?