BJTs are much more suitable than MOSFETs for driving low-power LEDs and similar devices from MCUs. MOSFETs are better for high-power applications because they can switch faster than BJTs, enabling them to use smaller inductors in switch-mode supplies, which increases efficiency.
The main division is between BJTs and FETs, with the big difference being the former are controlled with current and the latter with voltage.
If you're building small quantities of something and aren't very familiar with the various choices and how you can use the characteristics to advantage, it's probably simpler to stick mosly with MOSFETs. They tend to be more expensive than equivalent BJTs, but are conceptually easier to work with for beginners. If you get "logic level" MOSFETS, then it becomes particularly simple to drive them. You can drive a N channel low side switch directly from a microcontroller pin. IRLML2502 is a great little FET for this as long as you aren't exceeding 20V.
Once you get familiar with simple FETs, it's worth it to get used to how bipolars work too. Being different, they have the own advantages and disadvantages. Having to drive them with current may seem like a hassle, but can be a advantage too. They basically look like a diode accross the B-E junction, so this never goes very high in voltage. That means you can switch 100s of Volts or more from low voltage logic circuits. Since the B-E voltage is fixed at first approximation, it allows for topologies like emitter followers. You can use a FET in source follower configuration, but generally the characteristics aren't as good.
Another important difference is in full on switching behaviour. BJTs look like a fixed voltage source, usually 200mV or so at full saturation to as high as a Volt in high current cases. MOSFETs look more like a low resistance. This allows lower voltage accross the switch in most cases, which is one reason you see FETs in power switching applications so much. However, at high currents the fixed voltage of a BJT is lower than the current times the Rdson of the FET. This is especially true when the transistor has to be able to handle high voltages. BJT have generally better characteristics at high voltages, hence the existance of IGBTs. A IGBT is really a FET used to turn on a BJT, which then does the heavy lifting.
There are many many more things that could be said. I've listed only a few to get things started. The real answer would be a whole book, which I don't have time for.
Best Answer
I would not make the generalization that MOSFETs can handle higher current and are more power efficient. It very much depends on the application.
BJTs can be cheaper than FETs. This is especially true for high voltage switching where the much larger die area of FETs make them much more expensive. That's why you see IGBTs used for higher voltage motor controls. IGBTs still can't compete with FETs for switching speed (the minority carrier issue) but often 20kHz operation is fast enough.
BJTs also have some differences from FETs than can make them more convenient in some applications. BJTs start to turn on around 0.7V, where FETs mostly have much higher threshold voltages. For low supply voltages that can be useful.
For high frequency power conversion with small magnetics the BJT can't touch the FET and FETs are now used almost exclusively in these applications.