Differential Measurements, The Difference of T1-T2

When does a tech want to measure "the difference" between two signals either AC or DC? for what kind of circuits or applications

Please give examples

When does a tech want to measure the Differential of two inputs or two outputs? for what kind of circuits or applications

please give examples

Having worked in Radio transmission/ TV transmission and computer control of transmission kit for 40 years I have never had to do it.
Systems that work on balanced lines which require a differential amplifier are always designed to cope with unbalance with fully floating inputs, so it is immediately obvious if there is a fault in this area, i.e. earthing one input removes all output.
To adjust for common mode rejection, its easier to balance the common mode rather then to actually try to measure it because you need a better differential amplifier then the one you are checking.
It is some times require to measure a wave form across a component when BOTH ends of the component are at a high DC voltage, then a differential amplifier with a very high common mode rejection would be required, but this is unusual and a special buffer amplifier would have to be built/bought.
Frank