Writting of a tournament game Why is it compulsory to write a game in classic chess tournaments? If the purpose is the validity of the moves, then  why not in quick games?
There are several reasons having to write down the moves in standard chess games:


as a document, proof of the game and result (in some cases but not always the players also have to sign)
in order to confirm 50-move and 3-fold repetition rules
in order to count the number of moves which can be relevant for the time control (e.g. sometimes you have say 2 hours for the first 40 moves plus 1 hour for the next 20 moves plus 30 min for the rest)
to prevent cheating (players changing position of pieces, etc), though this is usually not much of an issue
for storage (so that you can later analyze your game or sometimes it might even be entered into a database)


With short time controls there is simply no time to write down the moves. Also with some longer time controls, it is sometimes allowed not to write down moves when the remaining time drops below some limit.

Just for information the technicalities on how to record (and how not to) moves are laid out in article 8 of the FIDE laws of chess