My 15 amp hair dryer has a 20 amp plug. Can I put a 15 amp plug on it?
Of course you can't.  The 20A plug was put there for a reason.  Believe me, the manufacturer would much rather have put a 15A plug on there to make it more widely usable - they didn't use a 20A plug to annoy you... It's because they can't use a 15A. 

The 80% rule also applies to portable appliances.  No portable appliance using a NEMA 15A plug can draw more than 1500 watts (which is presumptive that the working voltage is 125V). 

As George points out, modern bathroom receptacle circuits are supposed to be 20A.  There's an exception in Code that allows you to put any common-as-dirt 15A duplex receptacle on 20A circuits; that's just so you only have to carry 1 receptacle on the truck instead of two (x 4 colors x normal/Decora; 8 is enough!) It also allows builders to use the 50 cent outlets instead of the $5 ones. 

If your bathroom circuit is 20A breakered and wired with 12 AWG copper or 10 AWG aluminum wire, then you are at liberty to fit a 20A T-slot receptacle there.