To elaborate the above post a bit more (as well as other posts)
Advantages could be a higher available output voltage (= power) in high impedance headphones.
This is interesting for battery fed devices as one can (almost) quadruple the output power.
You get double the output voltage (as you basically have 2 amplifiers in counter-phase) and since P= U2/R.
For lower impedance headphones there might be no 'power' benefits when the current limits of each amplifier stage is reached.
Say each amp 'section' is limited to say 50mA and these 2 amp sections are in series this simply is the maximum current.
There cannot flow more current and since R is constant the power does not increase for low impedance headphones.
Then of course there are conditions where the current limit is higher in that case there is some increase in output power.
For desktop equipment creating enough output voltage and having enough current is no problem.
The most common problem is voltage rails which could be limited by the usual op-amps used.
There are plenty tricks for that + discrete designs.
Still we can find enough DT equipment sporting 'balanced' (as in a 4-wire headphone cable + 4 wire connector)
The reason for that is 'demand' and 'prestige' one could argue. Partly true perhaps.
Balanced is used in studio equipment to lower noise floors and above all influences from 'garbage inducing' surroundings.
Mains cables running alongside cables, lighting cables etc. over long distances could be in close proximity over certain lenghts.
What does this have to do with headphone cables... nothing really as these usually don't pick up something unless you have your phone against your cable and an EMC sensitive amplifier section in there.
So ... what could be the reason for a balanced cable for headphones other than have more power for high impedance headphones when a low power supply voltage rail is present.
The answer is simple.... Crosstalk.
No... not crossfeed. In fact it is closer to the opposite of crossfeed.
And then ONLY for 3-pin (and being cheap also 3-wire) headphone cables.
It is a bigger 'issue' for lower impedance headphones.
The reason is the ratio between the headphone impedance and cable+headphone plug+ internal wiring (in some cases) of the RETURN wire only.
The resistance of the signal wires does not matter here.
I can make a long post here with pictures but wrote it down in
this article already.
The other advantage of balanced cables for headphones, aside from more power (depends on situation) is improved stereo separation due to the lack of a common return path due to usage of cheap 3-wire cables.
The 4 wire 'balanced' signal thus has no common return path.
Those headphones that already have 4 wires (usually the dual entry headphones) only to combine the return wires in the 3.5mm or 6.3mm plug won't really benefit from 'balanced'. Well aside from the available output voltage increase when played VERY loud.
In short: common return wire gone in 3-wire cables and higher maximum output voltage (depends on circumstances) for 'balanced' operation.