What are the advantages and disadvantages of using a sanding screen vs sandpaper when refininshing a hardwood floor? When would you choose one over the other?
I have years of using both. I have no scientific data or anything like that. If I had big pieces of lint/whatever stuck in polyurethane then I would use a sanding screen. If I had a large area of issues I would use sandpaper. A screen will allow you to knock out things that are not flush with your surface while leaving your surface in tact - see @DA01's comment.
What I am not sure about the comment is for fine sanding. Yes screens are great when finishing off drywall mud when you have a few bumps but they are really good at any point where the area you actually need to sand is a small percentage of everything. Screens are for when there are imperfections littered over your surface, sandpaper for the whole surface (think divots and air holes).
I find that sandpaper is more precise, and i feel that screens last longer and pick up more mistakes. This is scientifically proven because I use screens a lot and my drywall pros never do.
For light wear (just surface scratches), a "screen and recoat" approach can restore the surface and appearance -- at least for poly; not sure about other varnishes. Deeper damage probably means a full refinishing.