In absolute terms, the increase in heat in the room is the same either way, as long as the burner is on for the same time and treating the room as an insulated system with none of the heat escaping.

However, the way the heat is liberated into the room is different: since the water has a high specific heat, it will slowly release the heat over a period of time while the burner will have a more intense effect over a shorter period. By contrast, the air in the room will transfer the extra heat to the walls, furniture, floor etc. So, without water the air temperature should go up quickly and fall off, while with water the temperature will rise more slowly, but last longer.

Finally, the boiling water will increase the humidity in the room, making its subjectivet effect greater. So, if you are measuring perceived effect, the room may seem warmer with the water, but would probably register higher on a thermometer without the water