Should have no appreciable effect on your electric bill. Over a whole year, you might notice a few dollars difference.
To me it is more of a marketing term with lower end receivers, because to truly deliver higher current, they need a big power supply. Yamaha's amps are very decent in their price ranges.
Some speakers require plenty of current, so a receiver that can deliver plenty of it will obviously be a benefit. There are many factors in determining if a receiver has enough power for your application; the speakers, room size, volume you typically listen at, etc...
4 ohm speakers operate at lower voltage and higher current than 8 ohm speakers (almost by definition). The amount of current the amp draws from the power outlet should be about the same, all other things including the speakers and sound level being equal.
Less efficient speakers (the acoustic suspension types tend to be less efficient) need more watts (and therefore more current) from the amp to achieve a given sound volume.
(Receivers with) class A power amplifiers draw more current in that they draw the same current (the maximum) whether playing loudly or softly. (Class B amps draw less power at low volume, but need careful design and better aging stability to not have/acquire distortion at low volume levels; class AB is a compromise between the two)