All it means for headphones to be "balanced" is that they lack a common wire to the amplifier ground.
Connecting balanced phones to an unbalanced amplifier makes NO difference at all, in any way.
On the other hand, it is not possible to connect unbalanced phones to a balanced amplifier.
So there is no comparison that can be made in that respect.
A balanced amplifier, all other things equal, has the SAME power output, and 6dB more gain.
But gain is NOT power. Twice the gain (6dB) just requires half the input for the same output.
And +6dB is NOT twice as loud. That would be more like +10dB.
How do you mysteriously double power? Where does it come from? It doesn't.
If you have no technical knowledge or background, then you are fodder for false information.