The main difference between these two game types is the length of turns - 1-12 minutes for Quick games, and 12-48 hours for Casual.
In casual game, you can make your movements at any time within the 12h/24h/48h period - then click 'end turn' to save your actions and either close the browser window or use the 'Go to Lobby' button in the new exit menu. Once the timer has expired or all players ended their turns, a new turn will start - at this point e-mail notifications will be sent out to players.
If you have valid e-mail in your profile and haven't disabled notifications in your settings, you will receive e-mail when new turn starts in casual games - very useful.
Exit the casual game by closing the browser window or by using the 'Go to lobby' button in the exit menu.