I never much got into either band, so I might have a more objective view on this. I'm honestly more inclined to grief the thread and just write "The Fall" or "Orange Juice" - which isn't really griefing at all, because despite what fans of either B or RS happen to believe, I'm pretty sure The Fall, Orange Juice, and any number of British groups decades hence might have done well enough on their own without the prior presence or 'influence' of John, Paul, Mick, or Keith. FWIW, the Chemical Brothers - an electronic act - might have leaned more on The Beatles than either The Fall or Orange Juice ever did, and that's to all three groups' credit.

That said, I've heard more Beatles tunes than I have Rolling Stones, but the more I hear of the Rolling Stones the more I like 'em. They're often gritty, at times shitty, and their music speaks to some primal stuff the Beatles couldn't ever touch. I say this as a guy who lives a lot inside his head, but there's no rule that says you've got to side with the indoor kids at every turn.

The Beatles are to music what Wes Anderson is to movies, except Wes Anderson's made me laugh and the Beatles never have. The Rolling Stones worked with Godard and clearly had a sense of humor.

I'm also more inclined to side with the duo bearing less Christian names than "John" and "Paul".

As embarrassing as Los Stones have been through the nineties and the naughties, John Lennon would without a doubt have topped them in that regard. Just, you know, imagine: John and Yoko, aging. Together. They might have played bizarre mom-and-pop jazz at the Clinton inaugural ball in '93, and we'd never have got that insane Fleetwood Mac reunion. Today, John Lennon might be reduced to comment on the latest Mumford & Sons, or he might have got into techno a la David Bowie in the late 90's. He'd probably attend a rave. He'd be wearing cardigans and harboring complex grudges against practically every artist who's ever touched a laptop, or alternately he'd be the one wielding a laptop and all the rockist dinosaurs who made him the legend that he is would flip out, and it would all be every bit as annoying as The Rolling Stones doing what they've done to your parents' ears ever since John died.

So I guess the answer's "neither", but if it's got to be binary then this outsider chooses the Rolling Stones, despite the fact that I'd have a hard time naming more than three of their songs from memory.