If the metal type is different, logically it follows that the various properties of the metal type will also be different (e.g., resistance, specific heat capacity, density i.e. mass divided by volume, and heat transfer rate). But for reasons that should be equally obvious, to be able to examine these differences everything else that isn't the metal type nor is among its properties must be kept the same, i.e., the thickness of the wire, the total length of the wire that also includes the length of the legs, the number of wraps, and the diameter of the coil all need to be identical. Changing the thickness and/or length etc. to compensate for any difference that occurs as a direct result from changing the metal type would technically be called cheating. Anyone who disagrees with this trivial fact can not be helped (pun intended).
That said, Kanthal A1 requires more power than SS 316L if you want the rampup speed of the coil to be identical. But if you are talking about the specific part of vapor production that occurs after the rampup is finished, i.e. when surface temperature is kept invariable, then no, the metal type has no impact on the amount of power required for that, due to the First Law Of Thermodynamics.