As I mentioned, all our gain stages are typically differential. In simple terms you can say each stage is its own phase splitter. I know some people think balanced circuit has two independent signal paths, but that is usually not so, there is really just one path which amplifies the difference.
The benefits of such processing are numerous and well known - all high quality electronic instrumentation is built that way - but fall well outside the scope of a small post.
BTW, one common misconception is to believe balanced circuits have lower noise - in fact their OWN noise is always higher, they just reject some of the external noise.
As far as why it is better to give the circuit a true and clean balanced signal even if it can develop one on its own - try this simple exercise. Face your friend and take his hands into yours. Now start turning his body left and right by pushing one hand, pulling the other. Go back and forth and see how easy it is to turn his body. Now drop one hand, and do it with just one. Not as good, heh? :)