Yes @alvinner2, for those of us who were there in 1964, the artificially-created rivalry (to sell teen mags, perhaps) was The Beatles vs. The Dave Clark Five. And to be completely honest and truthful, I vacillated between the two. In my Jr. High School’s 8th grade talent show in the late Spring of 1964, my little combo performed both "I’ll Get You" by The Beatles and "Glad All Over" by The DC5. But it was Brian Wilson and his Beach Boys who remained my favorite group. As for The DC5. they weren’t able to keep abreast of the rapid evolution that Pop music experienced in the mid-60’s, and were all but forgotten by 1966 (maybe even ’65).
Long-time record collectors are already aware of this, but in 1964 Vee Jay Records issued an album entitled The Beatles Vs. The Four Seasons (each LP side given to one of the two). When Capitol Records was in 1963 offered the first UK Beatles album for the U.S. market by Parlophone Records (a subsidiary of E.M.I.) they declined the offer! By the time their second album (what became named Meet The Beatles in the U.S.) was ready for release Capitol Records had seen the light ;-) .
We were all confused by both Meet The Beatles on Capitol Records and Introducing The Beatles on Vee Jay Records being in the racks at the same time, but bought both. The Beatles had greatly improved between the recording of the two albums, the first having been recorded in one single 14 hour session!
To capitalize (no pun intended ;-) on Beatlemania, Vee Jay came up with the idea of the Beatles Vs. The Four Seasons (their big Pop act) album. Even we young teenagers though it was ridiculous, and few bought it.
As for The Beatles vs. The Stones, this may be hard to believe, but I don’t own a single Stones album. Never liked ’em, and after reading the interview with Ry Cooder in Rolling Stone in 1970 or ’71 (in which he recounted what transpired when that band of wankers paid him to come over to London to record. Find the interview, you will never see "Keef" and Mick the same again.), they were dead to me. Having to watch Keith in the Chuck Berry documentary (Hail! Hail! Rock ’n’ Roll) he made was brutal. Witnessing his lecherous, leering behavior towards Norah Jones when they duetted at The Gram Parson Tribute Show in The Universal Amphitheater was a truly disgusting, horrifying sight. What a pig. And a grossly over-rated guitarist and songwriter. IMO, of course.