There seems to be a lot of differing opinions out there on what is best practice. A lot of people seem to be shying away from the wet sponge due to the thermal stress it imposes on the tip. I have personally always used the brass sponge only, but my Hakko station came with both a "wet" sponge and a brass sponge for cleaning.
What are the differences between the two cleaning devices? How should I decide which one to use given the type of work I do?