Why would that produce a better sound than turning the volume up in single-ended mode? Even with really dynamic material, I never need more than my old iPod classic can provide with these cans.
There are several differences between single ended and differential systems:
Differential, if the same amplifier as single ended and ignoring losses, has twice the voltage and four times the power output into the same load
Because of the above, each amplifier in differential is loaded with half the impedance of single ended.
The signal in differential no longer touches the ground, which can make design easier. The relatively high currents from a headphone can disrupt the signal integrity of the around low current parts of the circuit if the layout isn't perfect.
The above point also means the left and right do not share a return path in differential, either in the cable or the active circuitry.
The output impedance, unless specially increased in single ended, is often double in differential.
Any even order distortions in single ended can be cancelled in differential, while odd order remain similar.
However as some distortions can be load dependant, as the load in differential is half, those distortions can rise.
Most high end DACs have differential outputs, and is has benefits staying differential all the way to the transducer.
Differential is more noisy, but not double due to some noise being cancelled, and others being uncorrelated.
Differential costs more.
I could go on, but it is a mixed bag of features, which are more involved than a bit of output impedance and fancy cable.