Do you replace your summer mats with the winter ones, or just put the winter ones on top?

I have a 2005 Honda CR-V, and since getting the vehicle I have always just cleaned the interior -- including the mats -- well, before putting down the winter mats. Recently, I heard something that makes me wonder if this is not the right thing to do.

Someone told me that, actually, you should remove the summer mats, because, during the winter, melting snow can result in a great deal of moisture, and, if you have the winter mats sitting on top of the summer mats, it can create condensation (prevent the moisture from evaporating fast enough) and promote damage.

Should I actually be removing my summer mats, cleaning the floors well, and THEN laying down the winter mats? OR is it better to leave the summer mats under the winter one? If so, why?