What's the difference between MOSFETs and BJTs (from a circuit analysis perspective)?
From the design standpoint, the main and most evident difference is the base current: as Russel said, the bipolar is current-driven, which means that the current flowing into the Collector will be proportional to the current flowing in the Base (and the Emitter will output the sum for the KCL); the MOSFET instead, has a very high Gate impedance, and just putting a voltage higher than the treshold will activate it.

The bipolar transistor has a fairly constant current gain, \$ h_{FE} \$, which gives a linear response while the MOS has a quite complex response (quadratic with Vgs in saturation, depending on Vgs and Vds in "linear").

At the other hand, its fixed gain may be insufficient to use it as a switch, where a low-power input is used to turn on a high-current load: in that case the Darlington configuration (two cascaded BJTs) can help, but MOS doesn't have this problem because its current gain is virtually infinite (no Gate current as we said).

Another aspect that may be relevant is that the MOS, being controlled by charge in the Gate, doesn't like it to be floating (not connected): in that case, it's exposed to noise, and will result in an unpredictable behavior (possibly destructive). The BJT, requiring a base current, is more robust in this sense.

Usually BJTs also have a lower threshold (around 0.7 V vs 1+ V for the MOS) but this is very dependent on the device and does not always apply.