There are many good reason's for writing down your chess moves. Chess notation enables you to record your games for playback later. That day you played a superb game of chess... wouldn't you like to be able to play through it later and show your friends? Recording the game with chess notation enables you to study, replay, or show the game later!
You can also read and play through games of great players to learn and improve your own chess game! There is a wealth of materials both in book and digital form allowing you to study the chess moves of great chess players.
Chess competitions actually require recording of moves from the beginning scholastic levels to chess championship levels. The reason for this is that the game can be followed and recreated to any position in case there is a dispute. There are also chess rules based on the number of moves played and these can only be applied if recording has taken place.
Many players like to record their chess games and later enter those games into chess database software which allows them to further study and analyse their game. The chess software will analyse and point out mistakes or places where they could have played much better moves. It's a great way to improve your chess play. It's almost a secret weapon since a remarkably small percentage of chessplayers utilize this concept of chess improvement - and the tools are right there!