This post is not supposed to be very comprehensive when comparing the MOS with the BIP (bipolar) transistor. However, I thought it could be nice to have it outlined on a single sheet here. MOS stands for metal oxide semiconductor.
Inte picture below we find Lilienfeld vs Shockley. (Yes, yes, yes, one can argue about who did what, etc., but I will let them represent the two transistors). Lilienfeld represents a more square symbol and a “simpler” expression of the current as function of input voltage (in its desired operating region): a polynomial – the square of the input voltage, ube. Shockley presents an exponential function instead – the diode equation. I have sketched the currents as functions of the input voltage and we see that even though the square (MOS) is stronger in the beginning, the exponential quickly comes up to pace and produces higher currents.
The MOS layout is more compact compared to the bipolar. The MOS offers an infinite input resistance (well). The bipolar does not. In fact it must have an input current to operate as desired.
Considering the small signal schematics, the parameters gm (transconductance), gds/go (output conductance) and gp (input conductance) can all be derived as dependent on the current through the transistor. The higher current, the higher everything, sort of. Arguably, this implies that the gain is more or less with current.
At the bottom of the figure, we find the intrinsic gain of the transistors.
For the MOS it is the Early voltage over the effektive input voltage, i.e., gate-source voltage minus the threshold voltage. For the bipolar it is the Early voltage over the thermal voltage (~26 mV). These two gain expressions actually tell us that it is quite likely that the gain is higher for the bipolar than the MOS! (This can also be seen from the MOS transistor operating in the subthreshold region).
Why larger? It is hard to push down the MOS effective voltage to the required 52 mV to match the bipolar relying on the thermal voltage.