There are a lot of misconceptions around the questions of balanced and push-pull. Perhaps the simplest distinction is that push-pull is internal to the amplifying device, while balanced is a term correctly used with respect to interconnections.
However, in popular usage, the word balanced is often used to describe a push-pull circuit topology, especially a push-pull circuit which is push-pull at every stage (not just the output stage.
For the original poster, Bottlehead has so far found that single-ended zero-feedback internal circuits sound better to us than any others. We have made a very few products with feedback, at present the only ones are the Tode guitar amp (adjustable feedback to change the tone) and the Crack (the cathode follower output is a form of very local feedback). We have never made a push-pull topology.
For some of the technical misconceptions:
Push-pull can be Class A, Class B, or any compromise between them. That means the two devices can each handle anything from 50% to 100% of the signal swing; they are push-pull by virtue of being in opposite phase. (In fact, each device can handle less than 50% of the signal, though this creates a large amount of distortion. It's called Class C and is common in radio transmitters where the distortion products can be filtered out. It is still push-pull as long as there are two devices in opposite phase.)
A balanced line is two wires with equal impedances to ground, operated in such a way that the voltage difference between the wires carries the signal, and any voltage common to the two is rejected. It has nothing to do with equal and opposite; for example a balanced line can be driven with a voltage on only one of the wires, as long as the other wire has an identical impedance to ground. It's the rejection of the common signal, which eliminates many sources of hum and noise, that makes balanced connections especially quiet.
