That would entirely depend on the design of the thermostat. Most electronic ones would need the neutral just to power the electronics even if they don't switch the neutral.
Latest NEC and dwellings - 120 volt needs to be AFCI protected in most instances.
Which heats faster - well if you put in say 1000 watts of heat - won't matter 1000 watts is 1000 watts, just draws twice as many amps at 120 then at 240, still draws 1 kilowatt hour for every hour it runs.
Now if you put in 1000 watts of 120 volt rated heat cable and apply 240 volts to it - it will draw 4000 watts - will heat much faster - will likely burn out in no time also.