The competition between John and Paul is the engine that drove the Beatles to ever-higher feats of compositional glory. It could even be argued that, from Sgt Pepper onward, the Beatles became Paul's group, that if it were up to the others there wouldn't have been any more Beatles albums at all after Revolver. And yet they continued to put out masterpieces on a schedule of months (their record company was very unhappy with them for waiting a punishing 18 months between the albums Sgt Pepper and The White Album, with only Magical Mystery Tour, "All You Need Is Love," "Lady Madonna," "Hey Jude" and "Revolution" to sell in between -- sweet hopping Jesus, what a schedule). The fact that most bands these days can't be bothered to put out mediocre product on a schedule of decades says a lot for McCartney's professionalism and ability to inspire.
The competition between Lennon and McCartney's continued after the Beatles breakup, but took on a much uglier, detrimental turn. It would be nice if these two songwriting titans could bring themselves to compete with the other acts of the day, but the fact was that there were few others who could match their talents. Who is Lennon going to compete with, Bernie Taupin? Is McCartney going to worry about Steve Miller breathing down his neck?
So while it is unhelpful to compare apples and oranges (you know, why didn't McCartney start an Orange label for his records? That would be just like him), a Beatle fan in the 70s could not help but compare the products of their heroes, and Lennon and McCartney knew it. For the purposes of this piece, I'm going to begin the competition in 1970, even though Lennon started putting out albums before that; the competition ends in 1980 for obvious reasons.