I'm going through my music collection, and I've been listening to a lot of Beatles tracks lately. One thing that still comes up, to this day, almost 40 years since the Beatles broke up, is whether Lennon or McCartney was the better songwriter.
My simple answer is that they made each other better. Whether it was smiles and collaboration or "screw you, wait until you see what I have now", it seems pretty clear that they drove each other on. Competition seems to be a good thing for business, arts, most things really. It stops complacency and makes people focus on results and on improving them.
Having waffled enough, I'd have to say that I'm a McCartney guy. I don't dislike Lennon, I love many of his songs, but on the whole, I like McCartney's songs better. While I was exploring all this, I listened to Lennon's Greatest Hits (solo stuff), and found for the most part that the songs didn't grip me, and seemed a little dated. It might be a bit unfair, as he hasn't written much lately. Lennon put a lot more politics into his music, and while that may be commendable, it doesn't age as well. Songs like "Instant Karma" and "Power to the People" come off as very 70s. And some of his personal stuff is fine, but some of it I don't really click with.
McCartney seems to be more of the tunesmith, capable of writing the catchy hooks and melodies. I find that while everything he writes is not gold, he manages a few great songs with each album, and most of my favorite Bealtes songs are his.