Transistor Differential Amplifier Non-linearity
But we have the same situation as in CE amplifier. The voltage gain will as \$r_e\$ varies with collector current. 
This graph show us how differential voltage gain changes versus input signal change.



For the simulation purposes I used Rc = 7.5k and Iee = 1mA so re = 26mV/0.5mA = 52 
The differential voltage gain for Vin = 0V is 7.5K/52 = 145V/V

As you you can see the differential voltage gain is non-linear because \$r_e\$ varies with collector current (the circuit is symmetrical so we have mainly odd-order harmonics).   

And now if we add emitter degeneration resistor the gain drop but non linearity will also drop. 

The same condition (Rc = 7.5k; Iee = 1mA) but this time with emitter degeneration resistors RE = 220Ohm. So the gain Av = 7.5k/272 = 27.6V/V
.

The gain is now more "flat" vs input signal changes, therefore we have less non-linearity.