Actually, Puzo never much cared for 'The Godfather' book. He wrote it out of desparation, he was broke. In one of his other novels, I believe it was 'The Fortunate Pilgrim', there was a minor story about the mafia. His publisher said he should've added more about the mafia to his book. Puzo then began a rough outline of 'The Godfather' and got the publishers interested. 'The Fortunate Pilgrim' wasn't much of a success so Puzo was desparate.
So it wasn't so much that Puzo wanted to re-write 'The Godfather', he always said he wished he could've written it better.