Why AC socket polarity is important?
The wide slot is supposed to be the neutral, the narrow slot the hot. 

Neutral is nominally supposed to be near ground potential. However, there's no guarantee the receptacle was wired up correctly. 

If it is wired correctly, and if a correctly wired polarized plug is used, then the threads on something like an Edison-base light bulb will be near ground potential and there is less chance of an electrical shock than if the screw is at 120VAC with respect to ground. So, it's 'safer'. 



It's also backward-compatible with older non-polarized plugs that have two narrow blades, however newer plugs that are polarized are not compatible with older receptacles (barring the improper use of tin snips). 

Edit: As kabZX points out, when one side of the line is switched or (most importantly) when one is fused, it should be the hot side only.