This tool of the electronic engineering field appears today as one of the most widely used, from radars to medical applications, with billions of units around the world. The working principle seems rather easy to understand; as any amplifier the intended purpose is creating a signal with more amplitude, but in this case the point is to amplify only the difference between two signals.
And what's special about it? When the signal is traveling in environments that inherently have a degree of noise, there's the possibility of rejecting it. Noise in this environment would be called common mode signal, and when the mathematical operation of subtraction is applied to two signals that are the same the result is always zero. So, the greatness of it is when the noise carries other signal like radio waves or audio and the subtraction takes place. As you could guess, the noise is completely canceled! Almost like a miracle, the signal arrives clean and even ready to be considerably amplified.
For instance in audio an entire cable (conductor + insulator, sometimes meters and meters of it) has the sole purpose of intercepting noise. The other carries the audio signal and finally a shield for avoiding potential interference makes three cables. So the audio signal arrives with the noise that was picked somewhere, as the other cable with no signal. Subtract the two and the noise just disappear! Thank to this idea, audio recording and reproduction is quiet, so it makes its path through culture and art.
The differential amplifier also achieve great developments in the telecommunications' area. It's one of the best ideas for sure!