OK, so you have a couple of things going on here. I think you're trying to apply some concepts that relate to differential signalling at line level to your headphone case. Let me see if I can tease that apart. I'll start with line-level signalling between two pieces of equipment and treat the headphone case at the end.
Note that I use the terms single-ended = unbalanced and differential = balanced interchangeably.
Single-ended/unbalanced connections between pieces of equipment (say a DAC and an amplifier) suffer from one problem: Ground impedance. When you connect multiple boxes together, any difference in their ground potential will cause an error current to flow in the ground connection between the boxes. This sets up an error voltage (V = I*R) across the ground impedance. This is a problem because a single-ended/unbalanced input cares about the voltage difference between the signal conductor and ground, so when the ground potential is different between the two pieces of equipment due to the error voltage, the error voltage is treated as signal. For more detail, read Bruno Putzeys' article on the subject:
https://www.diyaudio.com/archive/bl...d1460406090-bruno-putzeys-micropre-g-word.pdf
The solution to this is to transmit the signal as a voltage difference between two conductors and leave the ground as a shield. An ideal differential/balanced input only cares about the voltage difference between the two wires in the signal pair. In reality, the voltage on the signal pair has to be within some voltage of ground (common-mode voltage range), but this range is typically determined by the power supply of the differential input, hence, on the order of ±15 V.
Because the differential input only cares about the voltage difference between the two wires in the signal pair, any voltage that is imposed equally on each wire in the pair will be rejected (common-mode rejection, CMRR). This is how a differential input rejects hum, for example. Not there are limits to this. A good differential input will have around 80-100 dB of CMRR, so if you somehow manage to get 1 V of hum injected on to the wires, you'll get 10-100 uV of hum on the output of the differential receiver.
Using a twisted pair further minimizes the hum by ensuring that any electric field that couples to the wire is imposed equally on both conductors. The twisted pair was figured out and patented by Alexander Bell back in 1881 for use in phone systems.
There are some more modern cable configurations that are even better than Bell's original twisted pair. Canare's Star Quad, for example. See demo here:
The bottom line is that differential signalling is a really good way to ensure good signal integrity, in particular in noisy environments. That's why you see differential signalling used in pro setups with long wire runs. It would not surprise me if residential setups are operating with the mains hum just below audible. At least I found subjective improvements in sound quality when I converted to differential signalling on all my gear.
Now for your headphone case: Marketeers have caught onto balanced = good, so now anything "high-end" has to be balanced. "Back in the day" when amplifiers had 1% distortion, you could lower the distortion considerably by converting to a balanced design. I have a few issues with this: 1) You only cancel the even order harmonics (those that many find pleasing to the ear) leaving the odd order harmonics (that many find to sound harsh) in the signal. That doesn't sound like a wise thing to do. 2) Most cancellation circuits, in my experience, end up making the performance
worse rather than better. This is especially true at the ultra-low distortion levels of modern opamps. 3) There is really no compelling technical reason to use cancellation schemes to lower the THD as modern parts already deliver THD that's orders of magnitude below audible (OPA1612 0.000015% anyone?)
Now before someone cries, "but THD isn't everything!" ... that's true. THD is not everything. But by improving the THD you usually end up with a lot of other improvements as well (IMD and multi-tone IMD spring to mind).
Where balanced
is relevant to the headphone case is for crosstalk or coupling between channels. If you share the ground connection between the two headphone drivers, you will have some of the left channel signal mixing with the right channel signal. Even the shared ground in a 1/4" plug can cause crosstalk. For example, I measure 115 dB channel separation in my
HP-1 when using the XLR output and about 95 dB when using the 1/4" output. The difference is due to the shared ground in the 1/4" plug. Now, both are below audible and I've never detected any difference in subjective listening tests, but it is definitely a measurable effect and is backed up by theory (Ohm's Law, to be specific). Both outputs on the
HP-1 are single-ended. In the XLR output, I just ground the (-) side of the XLR plug. I see no advantage of a balanced amplifier for reasons outlined above.
So to answer your question: If the only shared ground in the connection from the amp to your headphones is in the 1/4" plug, I doubt you'll notice any difference between 1/4" jack and XLR connections to the headphones.
Now, if you go out and spend $2k on a pair of Super Duper Balanced Bling headphone cables, I guarantee you that you will hear a difference. But that has to do with cognitive psychology and not any difference in the stimulus that reaches your ears. Just make sure you read the manufacturer's marketing babble so you know what to expect and which improvements to hear with the $2k bling cables.
I hope this answers your questions and furthers your understanding.
Tom