Is there any reason to use junction or Darlington transistors for power applications?
1) Power FET's and Darlingtons are two different animals. A BJT functions best as a linear device which is precisely CURRENT controlled. BJT's inherently have higher bandwidths than FET's and are generally cheaper for identical current carrying. In addition BJT's can make excellent and cheap constant current sources, making a simple but precise constant current source for sensitive current controlled devices like LED's. BJT's and particularly the Darlington configurations allow you to control precisely an output current in the 0-10A+ range with typically less than 2mA from an MCU with a simple current set resistor to the base connected to a microcontroller pin. 2) For precision using a PNP Darlington, the base current is referenced to ground, a microcontroller pin can still be used, the output is just turned low to ground the base resistor. If the main supply voltage varies, a current sense resistor needs to be used for feedback to compensate. Microcontroller pin currents do vary with sourcing/sinking ability and different MCU families will have different capabilities. A typical 5V AVR can source/sink up to 20-30mA/pin being TTL, and the SAM based arduino's like the DUE have two kinds of pin capabilities low and high current pins, high current pins which can only source 15mA/sink 9mA(low power CMOS) so keep this in mind if you're not using an op-amp as a buffer. 
 3) While BJT's are great at amplifying small signals with low distortion, and precisely controlling high currents, BJT's make for poor switches however because even if saturated, they still have Vce voltage drops over 2V, this means significant power dissipation at high currents, which means significant heat production. Even if you have a Darlington that can handle 20A before gain rolls off, having as little as 0.96A and ambient temp of 30C, you'll be at a junction temperature of 150C with no heat sink. 
 4) Power MOSFET's are nearly the opposite of BJT's in operation, they are great at being switches, but if not designed carefully, make for poor linear current control and amplifying devices. This has to do with the relatively large gate capacitances which limit the power FET's ability to have high bandwidths. Special gate driver IC's can handle the large charge/discharge currents when energizing a mosfet's gate capacitance at high frequencies but also increase project cost/complexity. 5)  Mosfets typically have much smaller "linear" regions than BJT's and have virtually zero "on" resistance as long as the Vgs conditions are met to drive the MOSFET into saturation. With "on" voltage drops Vds in the mV region, the only considerable power being dissipated is when the MOSFET is in transition from off to on and back. A typical power MOSFET can have continuous Id of 40A or more and not need a heatsink until you near half of that rating because the resistance of the MOSFET when on is usually in the milliohms region. With an ambient temp of 30C, a TO-220 case Mosfet with 0.01 Ohms RDSon (10 milliohms), would be able to dissipate the same 2.4W as a TO-220 based BJT with no heat sink but would be passing 15.49A without a heatsink at the same 150C junction temp! 
 6) Using a Darlington in a TO-220 case with an adequately sized heatsink can linearly control large currents precisely with just a few mA going/coming (NPN/PNP) to/from their bases. A Darlington can also be used to amplify small currents/signals accurately with very low distortion due to their larger "linear" regions (great for DC-RF precision power applications). Darlingtons are particularly well suited as a constant current source where output ripple from a switching supply would be a concern for your design. However this comes at a price with large voltage drops of 2V or more across the collector and emitter, leading to high power dissipations. BJT's are also prone to thermal runaway without considerate design being positive temperature coefficient devices. 
 7)  With careful design, a mosfet can be made to work in it's smaller "linear" region, but will dissipate similar power losses as a BJT while operating inside this "linear" region. However, MOSFETs are usually negative temperature coefficient devices (they are somewhat overcurrent protected). They are quite static sensitive devices (like all CMOS), so precautions should be taken and ESD equipment should be in place when handling FETs. 

BJT PROs:


relatively straightforward in usage, easy to control
cheap
require little support circuitry
DC to Radio frequency operation
not ESD sensitive, no ESD precautionary equipment needed to work with


BJT CONS:


Inefficient
have relatively high power dissipations (heatsinks are almost necessity)
Positive tempco could lead to thermal runaway and destroy the transistor
Need high wattage low value "ballast" resistors in order to parallel






Power MOSFET PROS:


Very low RDSon allows high current low power dissipation designs
gate current only occurs during gate capacitance charging/discharging
Suitable for high current density switching designs with small/no heatsinks
can be paralleled without "ballast" resistors (only for switching)
Logic level gate power MOSFETs with integrated gate charge pump drivers available
Most are negative temco devices 


Power MOSFET CONS:


Relatively large gate capacitance limits frequency from DC to ~10MHz
Require special gate drive IC's for high frequency/high power FETs
Highly ESD sensitive devices, requiring ESD precautionary equipment purchase
Logic level gate MOSFETs have fairly slow transition times Ton+Toff = avg ~44nS (22.7MHz near upper limit) - not really a con unless MCU freq > ~44MHz


Hopefully this can better clarify the suitability of BJT vs MOSFET choice for a given task.