A balanced output gives you exactly 6.02 dB more voltage, which is four times the power into the same load. That is the entire mechanism. Whether you get all of it depends on the amplifier's current limit, and there is a cost that almost nobody mentions.
A balanced output is two amplifiers driving opposite ends of the same driver. One swings positive while the other swings negative, so the voltage across the headphone doubles: +6.02 dB, four times the power. The driver cannot tell where the voltage came from. There is no separate sonic mechanism, no wider soundstage that arrives with the extra wire.
You only get the full four times when the amplifier is voltage-limited. Doubling the voltage also doubles the current demand, so into low-impedance headphones the current ceiling arrives first and the gain collapses. For a 2 V, 100 mA design the crossover sits at 40Ω: above it you get the full +6 dB, below it you get less — only +4.1 dB into 32Ω.
And the cost: balanced doubles the output impedance, because both amplifier halves contribute their own. Damping factor is halved. For a 300Ω dynamic headphone that is irrelevant; for a multi-driver IEM with a swinging impedance curve it can undo more than the extra power gains you. Check it against the one-eighth rule before assuming balanced is an upgrade.
Two more things worth knowing. Distortion usually rises slightly on balanced, because each half now sees an effective half-impedance load. And on many products the single-ended jack is derived from one leg of the balanced output, which means single-ended is by construction exactly 6 dB down — the balanced figure is the amplifier's real specification, not a bonus.
Never connect a passive splitter or a balanced-to-single-ended adapter that shorts one leg to ground. On a bridged output both legs are driven, and shorting one can destroy the amplifier.