Why would an aircraft perform an unpressurized takeoff? What conditions lead to this being beneficial?
Typically, the pressurization system is run by engine bleed air. Using that bleed air means you have less engine power available than is maximally available with the bleeds off. If you're on a runway that requires all your power, you do a no-bleeds takeoff, which means you have no pressurization. Shortly after takeoff you would turn the bleeds on.
Packs off doesn't mean that the cabin is unpressurized. The outflow valve(s) are closed and the cabin pressure remains steady at aerodrome level. This will give better engine thrust as mentioned above. Unpressurized cabin would mean outflow valve in open position and cabin pressure would be equal to ambient pressure. This is not allowed on commercial flights (at least JAA/EASA regulations). I flew one ferry flight unpressurized and almost popped my ears on takeoff because of the high climb rate.
One reason for an unpressurised takeoff would be if the aircraft's pressure hull is damaged, and it's making an unpressurised ferry flight to somewhere it can get repairs.