I don't think I've ever encountered anyone who loves one and loathes the other either. What I find most irritating is the tiresome comparison as though they were fairly equal heavyweights from their era. Let's get it crystal clear once and for all: There was a gigantic, immeasurable GULF between The Beatles and EVERY other group from the Sixties. I think it was Tom McGuinness from Manfred Mann who pointed out that The Beatles were emphatically head, shoulders, torso, groin, thighs and knees above ALL their peer groups (who should not have the cheek to dare call themselves "rivals").
So for me, although The Rolling Stones are good(ish) (but not as good as, say, The Kinks or The Hollies) to pretend they were a counterbalance to the fab four as some people do is just plain ridiculous. The Beatles transcended their era - and, indeed, music - by imprinting themselves into the permanent fabric of mainstream cultural life. Only Elvis Presley comes close to their universal, lasting impact.
People like Dylan and Hendrix were touchstones within their field but didn't influence shoe styles, haircuts, the shape of spectacles or vernacular speech like The Beatles did. The Rolling Stones certainly didn't. Ever. Before The Stones figured highly on most people's radar The Beatles were being played off against people like Gerry & The Pacemakers or The Four Seasons or The Dave Clark Five. The Stones were certainly not unique in being held up as the "alternative to The Beatles", just the most famous in a media concocted line of challengers who in reality posed no credible comparison.