A humid storage room loses more alcohol. A dry storage room loses more of the water volume.
The air inside a cask and the air outside both wants to have the same amount of water and alcohol, but the wood prevents it from flowing freely. Just a small amount passes the wood. If the concentration of water is (the numbers are chosen just to make the calculus easy) 100 in the cask and 10 outside, ten times as many water molecules want to leave the cask as there are molecules that want to enter. If the concentation would be 100 inside and 20 outside, only five times as many water molecules want to leave. You could say that in this relatively more humid outside environment there is more room for alcohol molecules to leave as well.