Saw the good answer to this question: Is there a difference between the two conductors in 120VAC wiring?
But I'm wondering as a followup, since the two primary prongs in an AC mains plug are different, why (in many plug configurations) are you allowed to insert the plug in either direction? Is there ever a case where the appliance cares which line is "hot" and which is "neutral"? Is there ever a case where it's important enough not to screw up?