I'm going to copy and paste the best answer I found when I was looking into the same thing a while back:

There are difference between the two.....obviously. I can only point out a few comes into my mind so chances are I am going to miss some: 1) First and most important is the input current. BJT is a current device that amplifier the base current by β. So it needs base current. MOSFET on the other hand as the name indicates.."Insulated gate". The gate is really a capacitor and the charge on the gate cause inversion at the channel right under the oxide insulator when turn on. So if you have very high impedance drive that cannot provide current, but only voltage, MOSFET will be a much better device to use. 2) BJT can go into saturation where both the BE and BC diode both turn on at the same time. When that happens, it will take some time ( into u sec.) to get out of saturation and turn off even if you remove the base drive. This is commonly called recovery time. MOSFET do not have that problem at all, as soon as the Vgs goes below the turn on voltage, the FET turns off. That's one important reason why digital IC today mostly are CMOS logic because the internal speed is so so much faster than TTL which is BJT. 3) When BJT goes into saturation, it still has about 0.3V across the collector emitter, MOSFET behave like a resistor when fully turned on. The bigger the MOSFET, the lower the Rds on resistance. That all sounded good, but there are few catches: 1) The capacitance of the input of MOSFET can be high if the transistor is large. A power MOSFET can have input capacitance in excess of 3nF. Driving MOSFET require strong drive to get speed. That's the reason why the processor can run GHz internally, but when the signals and bus interface to external logics, the speed is not very fast. Transistor for internal logic is very small, don't have much gate capacitance, so very little drive requirement is required. But when it has to drive external bus, much stronger drive is necessary to sink and source a few mA. The transistor has to be much bigger than those for internal logics and it need much strong pre drive. Sometimes, the internal logics has to be buffered a few stages before gaining enough drive for external bus. That really kill the speed. It is common for power amp to have BJT as a pre driver stage just to drive the big power MOSFET. 2) Small signal MOSFET has much higher 1/f noise that dominates the lower audio frequency range. The noise can extend all the way into MHz and beyond. Noise performance is almost inferior for regular speed op-amp, much worst for audio circuits. 3) the Gate source turn on voltage variation between MOSFET even on the same die is much greater than Vbe of BJT, temperature drift is much higher. If you want precision op-amp, BJT op-amp are mostly superior. 4) Because the gate source turn on voltage vary with temperature, propagation delay of logic change with temperature is much worst than TTL or ECL. For precision timing control, you cannot use CMOS logics....at least up to a few years ago. The major problem of MOSFET is the input capacitance that limit the usefulness. But in RF world, people make use of this. LDMOS is very popular in RF power amps. We always use matching network to get best power transfer. The matching network usually have a capacitor right before the input of the power transistor. A lot of RF power BJT are actually a hybrid where they put chip capacitors inside before the base. They have to put it inside because the lead inductance of the transistor become inductive at that frequency and create an inductor after the capacitor if it is outside. The input capacitance of the LDMOS is actually being used as that stage. This will reduce the cost tremendously as you don't need to put chip capacitors anymore. MOSFET and new group III and V transistor ( function either like MOSFET or JFET) really taking over BJT in microwave electronics. At microwave frequency, the noise of these devices are lower than BJT as it is beyond the 1/f noise range. All you have to do is using high pass filter to block the lower 1/f noise. At this frequency, BJT has a big disadvantage that it draws base current and thereby has current shot noise that can be bad. FETs has only thermal noise to deal with at microwave frequency. It is common to find FETs with noise figure below 2db or even 1db.

Reference https://www.physicsforums.com/threads/when-to-use-a-bjt-or-fet.640242/