There is absolutely no reason why balanced connections sound better.Balanced connections have the advantages that they tolerate ground loops and and more resistant to interference and that's about it. Neither of these advantages is significant in domestic circumstances. (In pro audio, you don't want to be chasing hum loops, and they can use very long runs, especially with mic cables, so the resistance to interference is important)
I am not saying I am correct as I am not technically savvy enough to claim otherwise but,,
Many years ago and not too long after Dr Zucker purchased Audiolab creating Tag Audio I auditioned a PA20R pre amp. To cut a long story short the dealer told me it had balanced outputs, I purchased balanced cables but when I set it up the be lanced outputs did not work. After discussing with the dealer Tag agreed to take back my pre amp and use it to develop the balanced outputs and provide to me at cost. Apparently the balanced output sockets were fitted as it was anticipated it would be a popular option in the German market.
The cost of the upgrade was approx £1500 but I had to pay approx £500. When I questioned this (spent a lot of time talking to someone technical at Tag) I was advised that developing and then fitting the output transformers etc was very expensive but the SQ improvement would be worth it.
This led me to believe that a proper balanced output is driven differently to your normal single needed outputs and better technically. I was told a lot of manufacturers add balanced outputs to their amps for wider appeal in foreign markets but these were not usually implemented correctly due to the increased manufacturing costs.
accept I could have been hoodwinked!!!