On guitar amps, what is the difference between the two "watts" figures?
Although both values are in watts, they are referring to two different power quantities.  The "15 watts" refers to the audio output power, i.e. the power delivered to the speaker. This is the power that generates the sound that you hear.  The "180 watts" listed on the back of the amplifier refers to the power drawn from the AC input line. This is the power supplied to the amplifier from the AC wall socket (120 volts at a frequency of 60 Hz).  It needs this power to run its internal electronics which produces the 15 watts of audio output power.  The AC input power will always exceed the audio output power since the amplifier cannot be 100% efficient.  In this case, its efficiency is pretty low (15 watts out for 180 watts in or an efficiency of 8.3%).