What is the difference between MOSFETs and BJTs
A brief comparison of mosfets and BJE transistors would include the following:

voltage drop


Transistors have a considerable voltage drop across the emitter and collector, it is in the range of 1-2v (and even higher for Darlington).
Mosfets on the other hand don't have a specified voltage drop but rather a drain-source ON resistance (Rds-on) which is in the milliOhm range and results in a low voltage drop.


base/gate drive


Transistors normally need a resistor to the base (a resistor in the emitter can have a similar effect) to limit the base-emitter current to safe levels. A general rule of thumb is that a transistor needs about 1/10 to 1/20 of the collector current driven to the base in order to be saturated and have a low Vce voltage drop (which is wanted for a transistor used as a switch). For a Darlington that current is considerably lower because of the high gain.   
Mosfets practically need no current to the gate for static operation (it's just a small amount of current while turning on/off) but need to drive the gate at a voltage level that is high enough (depends on the model) for the mosfet to turn fully on and have a low Rds-on.
When the mosfet is used for fast PWM switching (unlike static operation) then a proper driver (with sufficient current capacity) is usually required  and the reason is that the mosfet gate has a capacitance that has to be charged in order for the mosfet to be turned on and discharged for the mosfet to turn off, the faster you can charge and discharge the gate capacitance the faster the mosfet is going to switch between the on/off states spending as little time as possible in the state that is between these two where the dissipated power will be high (this result to overheat.)