First and second diagram here explain the basics of what “balanced” is about on headphones: https://www.headphones.com/pages/balanced-headphones-guide
The primary benefit is lower crosstalk because the common ground connection between the channels is removed. Will you hear the difference? Depends what system you’re coming from - a well-designed single-ended system will already give you lower crosstalk than is possible to hear (below -50/-60 dBr), so there should be nothing extra to hear when “going balanced”. I think only generic phones and computers still have crosstalk so high that you can hear restricted stereo separation because of it. Any decent external headphone amp should have no such problems.
Distortion is also not a clear benefit - sometimes it’s higher, sometimes it’s lower. Same story with noise - since you’re using more opamps instead of fewer, there’s no clear argument that “balanced” amps always have lower noise.
It has nothing to do with balanced lines and traditional balanced interconnects for long runs of cable in live/stage setups - your headphone cable is not long enough to suffer from EM noise in the audible band, so that kind of balanced connection would do nothing for you. As someone else noted, the headphone “balanced” connection does not actually balance any currents, so it’s a strictly wrong name for it, but it’s the name that stuck, and that’s what everyone calls it now. Technically it should be called dual-ground or differential drive or whatever, something to highlight that the channels have separate ground lines and (most often) are driven with a differential amp configuration.
Also it has nothing to do with higher power in the absolute sense - a single-ended amp that costs the same can give you the same max. power as a “balanced” amp. It’s just marketing when they say “double the power” - they mean relative to the same device’s SE output, not to other devices that cost the same. The only admendment to that is that these differential amps can give you more power than an SE amp if running from (and maxing out) the same power supply. So this might be relevant mostly in battery-powered devices, where you might get more power from something with the same battery size (but I haven’t seen any impressive examples of this so far).