Oh man, this post was way too technical for me to understand. I dont take it personally. Like I say in almost every post where I discuss any specific product, I am not an expert. I know nothing about headphones/amps and the technologies it uses. I only speak from experience and how certain things sound to me.
From my very basic knowledge of amps, I thought that by providing more power to the headphones, it makes it not only louder, but able to provide the sound signature that it was intended to make. I didn't realize it's adding it's own equalizer (though I do have to turn on bass boost on the E11 for my tastes).
But I don't mean to provide any false info to anybody, just give my own 2 cents as an amateur listener.
It's very fair, and I wouldn't ever wish to say I'm a better source of knowledge at all, since I've never even owned a dedicated headphone amp.
But, I've followed it up, and I've come down to two factors that can make an amplifier better at the same volume as the source playing the same material. One is output impedance of the amp. As you might know, the voice-coil of a speaker is a coil of wire that is moved by induced magnetism towards a permanent magnet. This movement will always induce current (moving metal closer to a magnet "induces" or makes current in the metal you're moving closer to the magnet), and that current will travel back to the amp. The thing is that any speaker is a resonant system, since the speaker diaphragms have mass, and their surrounds have stiffness. A lower output impedance on an amp will brake the unwanted resonant movement by putting a bigger current load on the current generator that the voice-coil is under movement, as said, it becomes a current generator due to the movement near the permanent magnet, and the higher load (because more current will flow with lower resistance) will brake this movement more efficiently. A good headphone amp can have output impedance under 1 Ohm, while an iPod has 5 Ohm.
Now, this is something the electrician in me understands right away.
The other argument is "The headphone amplifier improves the sound quality by increasing the amount of power available to move the transducer, increasing the control that the source has over just where the transducer is in space" which is close to the whole "providing more power without making it louder" - but the problem is that power is what moves the diaphragms in a speaker, and more power equals more movement, equals higher volume. "Driver control" makes no sense as a factor of power - it makes sense as a factor of output impedance as discussed above, but an amp can never assume anything on part of the headphones in an attempt to improve the headphones, since all headphones will have a certain differencing resonance, and harmonic distortion and intermodulation - all of which also variates with temperature to make matters worse.
All in all, amps are, all in all, a very simple electrical idea. An ideal op-amp, voltage source and some resistors is all you'd need to make the best amplifier in the world. The components we do put into an amplifier are never ideal, though, so it is obvious that the production value of an amp can be higher, and you can utilize different concepts for power amplification to make a good audio amplifier. There's nothing more to it. There's no need for any integrated circuits and power-controls or a tiny brain to control the input or output in any way. It's merely the result of transistors or other means of amplifications. I probably have the components to make a nice amp right here. This is why I'm baffled by a concept of "giving more power", because I know more power is a way of saying "louder".
Simply put - our source (iPhone, PC, PS3, amp) outputs voltage. Voltage squared over impedance is power. More power is only made by higher voltage, which makes the volume go up.
But - amps can also improve on intermodularity, total harmonic distortion, so there's no news that they can improve sound. And the driver control likely comes down to the dampening factor (helped by low output impedance of an amp). It's still a discussed point if this is a desirable effect or not, though.
