What the hell does fully balanced, quad differential, differential blah blah mean?
I'll take a shot at this. I don't know how much you know about circuit theory, so this might not be all that understandable.
Differential audio circuits are those where two independent circuits operate on two versions of the same signal, one being 180 degrees out of phase with the other - the mirror images of the signal. The two versions of the signal are created by what's called phase splitter circuitry at the sender, and when the signal is finally ready to get sent to a non-differential device, like a speaker, the two signals are combined to form a single signal again. The advantage of using differential circuits is that during the combining process at the receiver, the common noise and distortion in both signals are cancelled out to large degree, resulting in a final reconstructed signal with much lower noise and distortion than if you used a single path.
The disadvantage is that you have two circuits doing the work of one.
So, for example, if you have a low voltage signal, like the output of a microphone, and you have a long cable, by using differential signaling you can reduce the amount of noise and distortion introduced by the transmission of the signal on the cable. The noise might be due to interaction with other cables, or perhaps electrical fields from transformers or whatever.
In the output stages of amplifiers this technique is used to reduce noise and distortion, because the output transistors (or tubes) will highly amplify any noise or distortion created by the output stage circuits. The advantage is so great, the common mode rejection, as it's called, is often in the range of 70-100 decibels.
So, when an amplifier is referred to as a differential amplifier, it means that the audio signal is split, sent to two amplifying circuits, and then recombined at the output. In the high-end audio industry it seems like when manufacturers describe an amplifier as "differential" they are really only referring to the output stage.
Technically, balanced and differential are synonyms, but the high-end audio amplifier guys seem to using the term "balanced" to refer to circuits that are differential in every stage, and the signals are not combined until the final output stage. In fact, if you have a source component like a balanced pre-amp or processor, and you use XLR (balanced interconnects between the components, the combined signal paths of the source and the amp can be fully differential, and the common mode noise and distortion from the entire signal path will be reduced by the common mode rejection ratio.
Quad balanced or quad differential refers to taking a differential circuit, and then making a differential pair with another circuit just like it. So you have four circuits doing the work of one. You have cancellation at two levels. This sort of circuit is used in situations where extremely high CMRRs are needed, like high-speed digital circuits, but if you create a balanced output stage of an audio amplifier you get the same thing, because the output stage is already going to be differential.