Gain essentially is the drive capability of the output stage to power the next input stage. It is not like the attenuation of the signal.
Some power amplifiers are provided with input gain (especially in car audio and pro audio) so that it can be connected to varying types of input loads from low gain output stage to high gain output stage so that the power amplifier is not over driven or under driven.
Gain pots can be commonly seen in digital sound processors, power amplifiers, parametric equalisers and some other pro audio mixers.
Coming to how it changes the music, if not matched properly, you will either loose dynamics or introduce distortion. When properly matched it should not make change to the music.