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If you are already heating the space to maintain a constant temperature, and if both humidification methods result in the same eventual humidity, then the net energy used ends up the same.  Any energy saving by a more efficient humidifier just makes the heating system work a little harder.  Put another way, an inefficient humidifier uses more power than just required to vaporize the water.  That power ends up as heat, so the heating system has to produce less heat to maintain the same temperature.

However, there may still be big-picture differences in energy usage and cost.  If the heating system and humidifier use different power sources, like oil and electricity, then you can't so easily compare the overall energy usage.  The oil is burned and the resulting heat is used directly.  This process is fairly efficient for modern heating systems, in the 90% range.  If electricity is produced by burning oil in a power plant, producing heat, making steam, running turbines, which finally spin generators, then you're getting 40% of the energy in the oil to electricity on a good day.  There are further losses in the various steps to change the voltage, transmit the power, change the voltage back, before the humidifier consumes it.

Time of day can make a big difference too.  Is this during the day when lots of solar-generated power is being dumped onto the grid, in the middle of the night when mostly just the nuke baseload plants are running, during dinner time when the peaking plants are running?

The type of heating system also matters.  If it's a resistive space heater in a single room, then the electricity used is the amount of heating power directly.  If it's a heat pump and it's not too cold outside, then electricity is only being used to move heat from outside to inside.  The resulting heating power can be significantly higher than the amount of electricity used.  Put another way, the heating system can appear to be over 100% efficient.  Of course the 40% efficiency of converting heat to electricity at the power plant still applies (if that's what sources your electricity), but the the end result can still be over unity.

So the answer is a big <i>"It depends"</i>.  Many of the particulars are hard to know, and they change hourly, so in general more efficient units are going to be a win most of the time.