I usually clean mine. While there may not be any serious issues, it is SO much easier to clean the cage before the new spider has been added. If I start out with fresh, clean substrate, it will be that much longer before I have to relocate the spider for a good house cleaning.
I clean mine out too. Generally just the top layer of substrate gets tossed (if it's a sling to small juvenile) If it's an adult then whole cage gets cleaned (especially NW with all those hairs they lay down)
I clean out and refresh the enclosure too. When I move them I try to bring something like a piece of cork from the old one with them. Probably just a habit I have from the reptiles though. I doubt Ts are sentimental like that
I would definitely at least remove the old webbing, if not the new T may feel stressed if it's introduced straight into the previous Ts home. They surely realize they are intruding on another's territory, even if the other spider isn't present.
I always change out the old sub and webbing so they can start fresh and establish their own territories, plus substrate is not expensive so you don't really need to re use unless it's for the same T going into a different enclosure.