In my opinion, in the beginning Lennon was the driving force without a doubt; he put The Beatles together. Towards the end it was McCartney.
During their time together there was positive competition within The Beatles, hence some of the best songs of all time. Also, George Harrison provided some positive competitiion to both John and Paul.
After they split there was negative competition. Decent songs but nothing to match what they did when they were in The Beatles.
When they, sort of, made up, there was little competition between the two and, in my opinion, they produced nothing of the standard that they were capable of.
They were never anywhere near as good outside The Beatles as when they where together; they needed each other.