As usual, rules of thumb and knee-jerk answers can be misleading, especially when forgetting the constraints in the original guidelines that were then dumbed down to make "rules" for the non-thinking.
FETs are voltage-controlled, and BJTs are current-controlled. This alone leads to a whole set of tradeoffs between the two devices, having nothing to do with operating voltage, current, or power.
Both devices are capable of handling about the same power. Power dissipation is mostly a function of the package, and both devices are available in similar packages.
The advantages for voltage versus current control are not as simple and one-sided as others here would have you believe. The voltage control of FETs requires essentially no power to keep them in a particular state, but that ignores both the control circuitry and that changing the state often is necessary in many applications. A FET gate looks mostly like a capacitor to the driving circuit, so it takes current to change the voltage. That together with a typical 12 V gate swing over the full on/off range can lead to significant current and power. For example, let's say the total effective gate charge is 50 nC, and the FET is switched at 100 kHz (every 10 µs). That comes out to 5 mA at 12 V, or 60 mW. That's the same total control power into the device as a BJT with 80 mA drive at 750 mV. There are other concerns beyond these for driving FETs and BJTs, but I'm trying to point out that it's nowhere near as simple as "FETs take no power to drive".
In linear applications, the more predictable B-E voltage of a BJT can be advantageous over the D-S voltage of FETs. Constructs like emitter-follower generally have better characteristics than FET source-followers. Since BJT are both current in and out devices, they can be cascaded in ways that don't apply to FETs, like darlington pairs, or combined NPN-PNP devices. Another advantage of BJTs is the much lower voltage required to control them. You can control high current and high voltage BJTs with typical logic level voltages (3.3-5 V), which isn't possible with FETs.
Of course the voltage control and even the larger voltage control range of FETs can be advantages too. I'm not trying to make it sound like BJTs are better, just trying to point out some ways they can be more advantageous since the knee jerkers here seem to have decided FETs are "better" in broad classes of applications. FETs and BJTs are fundamentally different, so there are going to be various applications where one provides advantages over the other.
High current switching with low to medium voltage is one example where FETs are often used despite the generally more complex drive circuitry. This is because power FETs look like a low resistance when on, which can be 10s to single mΩ depending on how much money you are willing to spend. BJTs on the other hand look like a fixed voltage of maybe 200 mV to several times that, depending on how hard they are being pushed. At 10 A, for example, a 20 mΩ FET will have 200 mV drop, whereas a BJT will probably drop 2 to 3 times that.
FETs can also be more easily paralleled in high power applications because their on resistance goes up with temperature, unlike the BJT saturation voltage, which goes down with temperature.
For both BJTs and FETs, other characteristics become less desirable as the maximum voltage goes up. However, this happens more slowly with BJTs, so that above a few 100 volts, BJTs start looking like a good deal for power switching. In fact, this has given rise to the IGBT, which is FET and BJT working together. The FET is used to turn on the BJT, so doesn't need to handle as much current. The BJT then does the heavy lifting of switching the current and dissipating the power.
Again, different devices will have different tradeoffs, and devices as complex as transistors don't fall neatly into simple categories that lend themselves to rules of thumb. There is no substitute for actually understand what it going on, then weighing the tradeoffs for your particular application carefully to decide what parts to use.