I saw no need to be redundant and comment the same way in two threads, so I'll change it up just a bit and say why I picked The Beatles.
When it comes to live music and popularity, nobody compares to The Beatles. What they were able to do live and on the charts is unparalleled.
When it comes to songwriting, nobody has the diversity that The Beatles have. Everything from something like a love song like "Hold Me Tight" to complex classical arrangements like in "A Day In Life," The Beatles had it.
The cultural impact Binky points out can't be denied. What started as a deluge with Elvis became a downright avalanche of epic proportions with The Beatles. Especially w/r/t teenage girls, fandom, and a little uncomfortable biology along the way.
When it comes to recorded music, I think what they did sonically with music and recording far outpaced the Stones. I'm talking the techincal aspects of it. From a brief snippet by Listmania:
" 6. Studio Techniques
This item could almost be a separate list in and of itself. The Beatles (and their recording engineers) either pioneered or popularized Artificial Double Tracking (ADT), back masking, tuned feedback, spliced audio loops, distortion, equalization, stereo effects, multi-tracking (overdubbing), compression, phase shifting, and innovative “microphoning.” Although the Beatles are not credited with the invention of most of these studio tricks, they were responsible for directly inspiring countless musical acts that were desperate to copy their unique sounds."
Just unreal and so influential. This little comment, by the way, is inspired by krista4's Beatles threads and the amount of work and care that went into them. And a shout out to Doc. Oc. and his Stones countdown, which was also tons of work, well thought-out, and featured some of my favorite people on the board who are now unfortunately no longer with us (MoCS, wikkid). All top notch stuff.
But I had to pick the Beatles.