No, 240 V does not give you higher efficiency. A simple electric heater that uses a resistive wire to create heat is one of the few types of electrical devices that is 100% efficient, regardless of voltage. What 240 V gets you, as others above have said, is the POSSIBILITY of getting heating power four times a high as the SAME heater on 120 V. BUT all that really means is that the higher heating power will have to actually run for one-quarter of the time to produce the SAME amount of heat. The impact of that is that a higher-power unit can heat the space up from cold more quickly, AND if the lower-power operation is NOT enough when running full time, a higher-power one will give you more heating rate than needed.
OP, you say the circuit in this bathroom is a dedicated 120 V circuit, probably on a 15 A breaker. The only load currently is room lights (maybe - you did not mention this), a venting fan (likely 1 to 3 A) plus some heating unit associated with the fan. THAT heating unit will be a load you need to consider. Your best bet may be to DISconnect the heating system in the fan unit and rely solely on the new heater. You have estimated that new one needs to be 500 W, which can fit easily into the amp rating of the breaker, and certainly CAN be a 120V plug-in unit, not necessarily hard-wired. You may need to install an outlet for this, and it SHOULD be a GFCI type for use in a bathroom. That needs to be the same Amps rating (or slightly less) as the breaker feeding this room. I would suggest you get a heater unit of 1,000 W or so (more common that 500 W) with a thermostat built in so you have lots of excess capacity just in case, but it still fits into the electrical load limit of the room IF the heater built into the fan is disconnected.