Start the car implies simply turning the key and setting off on a journey. although 'get the car started' does mean the same thing, I think it could also imply that the driver has had difficulties in starting the engine.
Get the car started could also be an order. As I fly out of the bank with my gun in one hand and the loot in the other, I could be screaming "Get the car started!" (though for the sake of brevity, I'd probably say "Start the car!"