I play golf to relax. I have ambitions of getting better, but never to play competitively. I don't like to play for money, and find that if I am not keeping score, I don't have to justify taking the 3' gimmie for bogey, or moving the ball away from the tree to prevent the possability of damaging a club, etc. That being said, I find golf to be much more fun when I don't keep score. At the end of the round, I might reflect back on how many holes I shot par or better, or a certain fantastic shot. I just think it is way easier to forget the blow up holes when you don't have a written record of them.
Somebody once told me, how do you know if you are getting better if you don't keep score. To which I respond. Sure, I want to get better, but if I am having fun, and leave with a smile on my face, whats the difference. I know if I am getting better because of the confidence I have when I step up over a specific shot. I know I am getting better because at the end of the round, I have more memories of good shots, more memories of the sound you hear when the ball drops to the bottom of the cup on a 6' putt to get par / birdie, the comment your playing partner makes when you hit a nice golf shot, etc.