As others have stated, the voltage of the appliance/circuit has no bearing on the size (gauge) of the wire. Voltage dictates the quality of the insulation of a wire and most (power) wire we encounter will be rated for 600 Volts.
The gauge should be primarily selected by determining the current draw - in Amperes - of all the devices to be connected to the circuit...
Amps = Watts / Volts
TotalAmps = [Device1(Watts) + Device2(Watts) + Device3(Watts)] / CircuitVolts
TotalAmps = Device1(Amps) + Device2(Amps) + [Device3(Watts)/CircuitVoltage]
... and then referencing a chart which can be traced back to the NEC recommendations. Herein lies the problem. There is no one official NEC "if X then Y" chart for all situations. The actual NEC charts are for engineers/contractors to reference when designing for an application and are not very easy reads. Here is what the NEC has to say: http://www.fs.fed.us/database/acad/elec/greenbook/3_basicdesigns.pdf
Fun, right? What we normies have to do is rely on charts that interpret those recommendations and those charts vary - sometimes wildly - in easy of readabilty. Compare my favorite chart http://www.cerrowire.com/ampacity-charts to this one http://www.usawire-cable.com/pdfs/nec%20ampacities.pdf They are both technically accurate from a rule of thumb basis but the latter requires more in depth evaluation such as Note4 which indicates a derating of the wire's maximum ampacity if the conduit fill (number of wires in the raceway/cable) is greater than 3.
Amperes is not the only factor for wire size, but we are working with rule of thumb here. The other MAJOR factors that contribute to selection are (A) the wiring installation application type (THHM, UF, etc...) and temp ratings, (B) the length of the circuit feeder which increases resistance, voltage losses and ultimately, unacceptable heating of the feeder wire and notably it's connections, (C) single- vs multi-phase applications (we are only concerned with single in household systems), (D) whether the load is inductive or not (big motor/compressor in the appliance?) and a couple of other more obscure factors we won't address here.
Item (A) in residential applications is typically NM/NMC class wiring for Romex-style, THWM for BX or conduit-style wiring and UF for cable buried in your yard. Item (B) is actually quite important. If the wiring run is very long, the resistance of the wire (all wire is resistive to a degree) and thusly the temperature of the wire will increase. If that temperature rises above a wire's insulation rating, it could melt causing a short or worst case, start a fire in the surrounding building materials. This is where my second favorite chart comes in: http://www.cerrowire.com/voltage-drop-table
EDIT: longnecks's top-rated answer above is a better explanation of temperature's effects on circuits especially regarding the wire/fixture interface where most fires begin.
Knowing what we do now after referencing those two charts from Cerro we can answer:
If I dedicate a circuit to this appliance, the length of which is
about 50' (including the return), will 14 AWG be sufficient? Why or
with YES because you indicated that the device will be the only one on the circuit and because the run is actually 25' by the definition of the rules which do not calculate the total length of WIRE, rather the length of the CIRCUIT which is comprised of both conductors. In 240 land, there is no Return or Neutral. This allows 240 circuits to often use a gauge that would seem to be too small! In 120 land, the neutral of a given circuit is allowed to be (and nearly always) shared amongst the numerous branches of said circuit which introduces some derating. But mostly because circuits of a higher voltage introduce less voltage drop then an equivalent circuit at a lower voltage.
E = R * I ... where
E = voltage drop (volts,
R = electrical resistance (ohms, Ω);
I = current (amps,
A) This is not intuitive because the supply voltage is not used in the calculation. However, if you have two loads which are both rated at 2400 Watts, one of which runs at 120V and another at 240V, the former will draw 20 amps, the latter 10. Half the current draw will introduce only half the voltage drop, reducing that element of the calculation for a wire's guage.
It should be noted, that the answer would still be "YES, 14awg will do" if the run were actually 50' according to the Cerro charts....BUT just on the edge. After browsing a few other charts that are popular, some indicate 12awg, others 14awg. YMMV. That's why we have the really in depth NEC findings to fall back on and take into account EVERY factor.
I realize 12 AWG wire may be more difficult to work with, and it costs
more than 14 AWG. Beyond these, are there reasons not to do so? Are
there any advantages?
The answer is a judgement call for the contractor/homeowner. Take this example: I'm running a new 240 circuit for a new window air conditioner. The unit I can fit in the window opening can be handled by a 14awg/15amp circuit BUT is right near the maximum rating. Suppose the unit is barely able to meet my cooling needs and suddenly, the market introduces a higher BTU output unit that fits in the opening but it's going to require a 12awg/20amp circuit. This would be a future-proofing judgement call.
And remember the most important thing: Your local building codes supercede NEC's. If it's your property, the work you do along the way may impact your ability to sell the property down the road.
Hope I've answered all your questions. Disclaimer: I do not work for Cerro cable, just a tired old HVAC/R pro that deals with a lot of crappy wiring, residential and commercial. And the links are munged because this site only allows two links for noobs.