What is the purpose of using a differential amplifier? (Common-mode rejection ratio: CMRR)

Differential amplifiers are used mainly to suppress noise.
Noise consists of typical differential noise and common-mode noise, of which the latter can easily be suppressed with an op-amp.
There are two main causes of common-mode noise:

1. Noise is generated in the wires and cables, due to electromagnetic induction, etc., and it causes a difference in potential (i.e., noise) between the signal source ground and the circuit ground.
2. Current flowing into the ground of a circuit from another circuit causes a ground potential rise (noise).
   

In either case, the ground potential, a reference for a circuit, fluctuates because of noise. It is difficult to remove common-mode noise with typical filters. Differential amplifiers are used as a means of suppressing common-mode noise.
The op-amp configures this differential amplifier as the main circuit. The symbol shown below represents a differential amplifier. It has two inputs: V_IN(+) and V_IN(-). The output voltage is equal to a difference in voltage between the two inputs multiplied by the amp’s gain (A_V):
V_OUT=A_V{V_IN(+) - V_IN(-)}
Suppose that common-mode noise (v_noise) is superimposed on the differential inputs. Then,
V_IN(+)‘=V_IN(+) + V_noise
V_IN(-)‘=V_IN(-) + V_noise
Hence, the output is expressed as follows. This indicates that the differential amplifier
cancels out common-mode noise:
V_OUT=A_V[{V_IN(+) + V_noise} - {V_IN(-) + V_noise}]
=A_V{V_IN(+) - V_IN(-)}

The common-mode rejection ratio (CMRR) is specified as one of the electrical characteristics of an op-amp.