Unless the load on the heater is constant you won't get the 1 deg tolerance. There is a difference in temperature, between the point where the switch goes on - then goes off. I think "differential" is the term used by some manufacturers. A mechanical thermostat is unlikely to have a differential small enough to give you the control you want.
If you were to reduce the power to the heater to a point where the heat input matched the load you would need no control at all. Unless you have one incredibly large rheostat you can't control your 1500 watt heater that way. Even if you reduce the power - if the thermostat has to cylce to maintain the temp then the swing in the room will still be significant.