AG6QR: If the limitation is just power, why is it rated as current?
Still, why are cables rated for amps while resistors are rated for watts?
In the case of a cable, the limitation isn't just power, it's power per unit length. Longer cables can dissipate more power, because their surface area scales linearly with their length. Note that their resistance scales linearly with length, as well. These two factors cancel out.
As an example, imagine cable A is 100 meters long, with a resistance of 1 ohm, capable of 5 amps. The power dissipated is I^2*R, or 25 watts.
Cable B is made of the same kind of wire, but it's 200 meters long. Its resistance is double the resistance of cable A, and its power dissipation capability is also double. It has a resistance of 2 ohms. Again, at 5 amps, the power dissipated is I^2*R, or 50 watts. Twice the power in twice the length produces the same amount of heating above ambient. The same current was delivered in each case.
Because the wire makers don't know how long you're going to cut their wires when you make a cable, it's easier for everyone if they rate the wires in amps, rather than in power per unit length.
Obviously, a given resistor is of fixed size, as opposed to a wire which is cut to length by the installer. That's why resistors can be rated in watts. In fact, the power dissipation capability of a resistor is largely due to the package size and thermal characteristics, so resistors of widely differing resistance values, but the same size package and type of construction, will have nearly the same power handling capability. That's another reason resistors are rated in watts and not by current -- the same power rating can apply to all available resistance values across the series.