They are one and the same, the vapor is the gas. The air contains water molecures which come out of the gasseous state and condense on the cool surface as the air is cooled, causing the relative humidity of the cooled air to reach 100% (dew point) and the condensation forms. This is the function of a common dehumidifier. Humid air is drawn over cooling elements, which causes the air to cool and deposit some of its water molcules on the cooling surfaces (condensate), which is collected in a drainage pan, The expelled air will have less water molecules per unit volume.
I think you are getting hung up on the term "relative humidity"..It just means at a given temperature and pressure, how much water the air is holding compared to how much it CAN hold before reaching dew point.. As temperature increases and pressure decreases, the air can hold more water. As temperature decreases, and pressure increases, the air can hold less air.
If the temperature in a room is 75 and the air is half way to dew point in the amount of water it can hold, the relative humidity is 50%. If you warm up the room to 90 degrees, the air will be able to hold more water, so the relative humidity will decrease, even though the amount of water in the air has not changed.
This is why a hair dryer works. The heated air now has a low relative humidity, and can hold more water (think thirsty air) and can evaporate the water more readily (the oposite of a dehumidifier, you are actively putting water vapor in the air.
