Differentially Balanced kit means 2 totally independent channels (not touching each other), including 2 separate power supplies (one for each channel/L/R), but often a single PS is substituted ...
That's not at all how a differentially balanced circuit works. A differentially balanced circuit uses both an inverting and non-inverting signal at its input, then amplifies the voltage difference between the two signals to produce the output.
Depending on the design of the balanced circuit, there can be substantial benefits to using it in balanced mode, rather than SE. For example, with ARC balanced preamps, you'll get 6 dB more level using it in balanced mode. That means 6 dB improved s/n.