Differential amplifier and large signal analysis
Qualitative/intuitive answer (without exact calculation):

If the current source in the common emitter path is non-ideal (parallel resistor RE), there are two effects:


the common mode gain Acm=-gmRc/(1+2gmRE) has a finite value, 
the maximum possible current through the amplifying transistors is reduced.


As a consequence, in the transfer curves the maximum possible current is reduced and the minimum possible output voltage is enlarged. More than that, due to the finite common mode gain (negative sign) the crossing of the curves at Vin1=Vin2 (Vd=0) is somewhat shifted. Hence, the curves are not fully symmetrical anymore.