They do "care". It's not a huge thing, but it does matter. A water-cooled turbo has a water jacket that keeps the turbo from losing the heat if there's no water. A "dry" turbo has no jacket and they have little heat sinks to increase the surface area.
Bottom line: A water-cooled turbo will last longer than a dry one, but it won't last as long as a dry one if you don't run the water.