IMO, this has to do with how a team starts building attacks.
When counter box ticked, players will seek for an opportunity to launch a quick attack. If a player with ball (def or DM) sees such an opportunity, he will try to use it, otherwise he will wait for AI to come at him or pass ball short. Depending on mentality (Risk-Reward setting) and other settings, a player will try to start counter attack more often (higher mentality, quick tempo) or stay with the ball waiting. All this applies mainly to the early stage of attack - once ball is near opposition area there is no big difference how team plays with or without counter box.
Without counter-attack box ticked, a team moves ball forward more gradually (depends on other settings, of course). Usually you see less fast breaks as well as less dayling on ball by defs.
IMO, one of the reasons why in-game tip says "...counter-attack should be quick and direct" is because quick tempo implies that players should move ball fast, which reduces time a player keeps ball. Direct passing is not so necessary in this case,
IMO, as quick attack can be launched after series of short passes if AI actually comes after ball. Basically, to start quick attack you may not need lots of passing at all - drop the ball to pacey winger who is asked run with ball often - you get counter attack.