Why are they called quasi neutral and not exactly neutral? What happens which make them that? In the analysis we have done to derive the I-V characteristics , we always maintained neutrality where excess electron concentration=excess hole concentration in any of the quasi neutral region.
The concept of quasi neutrality as opposed to perfect neutrality comes from the fact the in a real PN junction, the space charge doesn't abruptly stop outside of the depletion region especially for lightly doped junctions. As the next best approximation to the abrupt junction in which bulk is neutral, in the graded junction there is appreciable space charge on either side of the graded junction. Thus calling the p and n sides neutral isn't completely correct.
Why does it make a difference? Well, the presence of space charge governs the junction field and thereby everything else. So there is an effect on the accuracy of calculations.
You may find Section 5.6.4, Solid State Electronic Devices by Streetman & Banerjee helpful in this regard.
To my understanding, this terminology is used to stress the fact that, although the medium is composed of charged particles, charge separation is only seen in small scales.
In larger scales, the medium appears neutral, due to collective effects that tend to cancel out any non-zero concentration of positive or negative charge, "shielding" (or "screening") them from the outside world.
The characteristic length of "shielding" depends on the temperature and the density of the carriers. It is dubbed Debye length by the plasma physicists or Thomas–Fermi screening length by the solid state physicicists.