Why hair dryer work with bed room outlet but not work with gfci outlet at bath room?

It could be that the bathroom circuit simply has more on it so the hair dryer is pushing the circuit beyond its limit. In a typical bathroom, properly wired, this would not be the case. But if there are lights on the circuit other than in the bathroom (it would take a lot of lights for this to be a problem), or if there is a source of heat in the bathroom that is on, this may be where your problem is.