Differential Balanced Sound Quality

You'll get higher gain and improved S/N when running a differential amplifier in balanced mode, and you'll also get the benefit of CMR.

Balanced connections are all about eliminating external noise that gets picked up in cables etc. They also pretty much deal with ground loops as well. So if either of those things are a problem in your system then using balanced connections is effective way of dealing with them. You also get a 'free' 6dB of gain when the balanced signals are mixed back to one signal... but I don't think gain is in short supply with modern equipment.
The bit that a lot of people are unaware of is that balanced inputs are actually noisier than single ended (RCAs etc.) in that they utilise relatively high value resistors that introduce their own (johnson) noise into the signal.
So if you don't have problems with interference or ground loops then single ended will be quieter. It's really down to your own system as to which is better.

When I read your first sentence I thought you might have been talking about bridged amplifiers vs single ended... that's another subject if you did.

I've read where running a true balanced (differential) amplifier as such sounds much better than running it single ended (I'm assuming the same amp has both balanced and single ended inputs here).

That's the story. The one time I was able to do a valid comparison that is sure not the way it went. Dealer wanted to show off his Krell and how great it was balanced. So I made him let me hear it with the same interconnect both ways. Obviously not the same RCA/XLR but the same brand and level interconnect. As close as close can get.  

With balanced there definitely was a blacker background and...  that was it. The sound had slightly more of a hard edge to it. Just the kind of thing a lot of guys confuse with detail. 

Anyway, it is all beside the point. Because it is so obviously the wrong way to go. The number one factor in quality sound is quality parts. Going balanced requires twice as many parts. A whole duplicate circuit. So even if there is some advantage, it would have to be twice as good. And it's not. Nowhere even close. At best a slight improvement. Which you paid a hefty price tag to achieve. Bad move.

Balanced is in my opinion a misuse of technology. Balanced is totally designed for professional use. Runs of hundreds of meters. Not tens of feet. Components with hundreds of connections frequently being moved around. Home systems have a few and they tend to not be moved around much. On and on it goes. Every design attribute of balanced has the professional in mind. Home systems need locking connections? I don't think so. 

Good choice though if your system is the Kingdome. 

You’ll find the thoughts on this very polarizing, and in reality, it just depends. My current system is all single ended, with the exception of my phono pre out. I run that balanced for several reasons. I’m not a fan of preamp to amp balanced connections. I’ve always preferred single ended overall. YMMV