A balanced audio cable has three parts to it. The signal is carried between two conductors, but neither one is ground/shield. On a 1/4" TRS (Tip, Ring Sleeve) the audio plus and minus will be carried on the tip and ring of the connector, with the sleeve being connected to ground.
The balanced input only see the difference voltage across the audio plus and minus connections. The two wires are twisted and this helps to reject noise. Outside noise does not get amplified, or in practice is very well attenuated because the noise is the same on both audio lines and the next piece of equipment only cares about the difference on the two lines.
The cable shield does not carry any signal. It does act as a shield though to further reduce incoming noise from outside of the cable. Often, only one end of the cable shield will be connected to ground. They can be connected at both ends, but if you get some ground loop hum you can disconnect on end of the shield. The other end is still connected and still shields the entire cable.
If you look at a TS cable vs a TRS you can see how the ring will be shirted to ground. Thus causing the loss of 1/2 the signal level.