Ok, I thought of an idea for an analogy concerning headphone amping:
Imagine you're buying a car, and you get a new top of the line Ferrari.
This brand new car is top of the line, amazing, but if you put in biofuel into it because it's cheap, it will not perform the way it was made to perform, however, if you put in premium unleaded it would be performing as it was intended - at its full potential.
Both fuels make the car work reasonably enough, but the more expensive fuel uses that expensive new car you bought run as it should.
This is the same with headphones and amplification. Imagine the amplification to be the fuel of the car, an E7 or onboard soundcard is going to be biofuel, whilst a good quality amp matched to your sound tastes and power needs of the headphones would be the premium unleaded.
Hope this rather poor analogy helps!