Air can hold a certain amount of water vapor at a given temp. The higher the temp, the more water air can hold. Relative humidity is the amount of water vapor in the air divided by the amount the air can hold at that temperature.

As air cools, the RH will go up until it gets to 100%. Once air reaches 100% RH, if it cools any more, you get condensation (dew on grass). As the air warms up, it can hold more water (dew evaporates back up into the air).

With all of that said, you probably have the same amount of water in the air on both your warm side and cool side, but the RH will be different.