For the thought experiment in your question, the answer is: it depends on the temperature of the room!
If the room is cold, humidity will actually make it feel colder! This is because water vapor has a much higher heat capacity than dry air, meaning that it takes more heat to raise or lower its temperature. So a volume of air with a lot of water vapor can transfer a more heat to (or from) your skin than the same volume of dry air.
If the room is moderately warm (around 27-37°C, I think), it's close enough to your own body temperature that radiative heat transfer is minimal, so the main difference in perceived heat is due to the water vapor's effect on your body's ability to use evaporation (of sweat) to cool itself. This is what the other answers are talking about.
If the room is hot, heat capacity again plays the dominant role. A given volume of high-humidity air contains much more heat than an equal volume of dry air, so it is able to transfer a lot more heat to your body. This is why dry saunas are frequently operated in excess of 100°C, while a steam sauna at that temperature would be instantly scalding!
In a hot room there is also the issue of condensation. If your skin is at or below the dew point, water will condense out of the air onto you. Condensation is an exothermic phase change, so it will transfer additional heat to your skin.