Considering a Common Emitter configured transistor amplifier.
re = 25mV/Ic
For a fully bypassed emitter voltage gain, Av = Rc/re and the variation of re with collector current causes distortion because Ic varies as the output amplitude varies which varies re, varying the gain.
As the output amplitude swings positive, Ic reduces increasing re and reducing the gain. As the output amplitude swings negative, Ic increases, reducing re and increasing the gain the gain. Therefore positive going output excursions are reduced in amplitude and negative going output excursions are increased in amplitude.
For a partially bypassed emitter:
Voltage gain, Av = Rc/(re + RE1) where RE1 = the unbypassed emitter resistance.
So, in this configuration the variation in re with collector current becomes less significant as re is in series with RE1.
So, the advantage of a fully bypassed emitter is higher gain with the drawback that distortion is increased.
Typical Av (No emitter bypassing) = 2 to 10.
Typical Av (partially bypassed emitter) = 10 to 50.
Typical Av (Fully bypassed emitter) = 50 to 500 but with increased distortion.