The manager take care of the money and the monkeybuisness all around that, PR, etc,etc. The producer sets up the recordings and are "a part" of the recording. Helps out with the sound and "lead" the recordings towards diffrent paths. He is the man who pulls everything into place.
Many times the Producer also is the owner of the recording and get royalties for it as such, all deepending what it says in the contract, prior to the recordings.
I count Billy Preston as the "fifth" Beatle as he did most of the keyboards during the late years.
George Martin, was a Producer and even though he contributed to a lot of the Beatles sound, he was nothing more.