Like John T. said, it helps a lot to use the tool to get the smoothest finish you can by the time you are done turning. There is usually a particular grit with any set of sandpaper, where its much harder to introduce those deep scratches. For wood I usually use a set of wet/dry sandpapers I get from Woodcraft (I forget the brand), for resins I use either MicroMesh or Zona paper (or both).
I have found that the 320 grit or lower sandpaper will almost always leave deeper scratches that I then have to get rid of. Even with a light touch, sandpaper is a cutting tool, and after that initial few seconds of contact a lot of the grit will be ground down, but you sometimes have random remaining sharp granules that can introduce deeper scratches. I usually try to turn my blanks so that I can start at 400 or even 600 grit. With 400 grit there is still some potential for deeper scratches, but with 600 grit it pretty much never happens. If I can get a really smooth surface just with the tool, I start at 600.
If you find that you have deeper scratches, usually you have to back up the grits to the next one above the grit that causes the scratches, and work them out. Sometimes you may even need to back up to the actual grit that caused the scratches. It helps to check for deeper scratches every grit or two, and see if you need to do this, before continuing through the final grit.
Resins can suffer the same issues. Wet sanding can really help when sanding resins. Some times I'll start with 400 or 600 grit dry sandpaper with a light touch just to smooth out the shape if it is not perfect, then wet sand from there on. I used to use micromesh exclusively with resins, but I recently found Zona paper, which has been giving me a much, much better, shinier finish by the time I get down to the 1 micron grit. I then use a high grit plastic polish (I've used a number, I don't recall exactly which one I've been using lately as the label fell off the bottle, but it does wonders and gives me a very, very shiny, smooth finish.) I have always found that with resins, sandpapers alone (normal wet/dry, micromesh wet, zona paper wet) never quite seem to get me to a true scratch-free point. I've always had to use a polish (or a series of progressive polishes) to reduce the scratch size to "effectively invisible".