To prove when cheating or illegal moves occur.
To avoid "misunderstandings" that feel like cheating where an opponent "conveniently" benefits from the misunderstanding (basically cheating but not as clear).
If a result is marked incorrectly, and then submitted to the United States Chess Federation, it is easy to overturn the result if the player has a notation sheet of the game. There were 3 incorrect results posted in January 2013!
Notation is most important for improving. It makes you play better while you are notating. Of course there is a learning curve. First it is distracting because it is a new thing. With practice it becomes automatic like signing your own name. Notating being "distracting" is not a very good excuse for anyone unless that player has problems with writing in general.
Notating gives players a nice routine during the game and helps the player stay calm and focused.
After the game is over the player can review the game and find "leaks" in his game while learning new ideas.
Sent from my iPhone.