-On means he was wearing the vest.
-Covering means his vest was on his body, but he was not wearing it (e.g. it was tied around his waist or he was carrying it over his shoulder etc.)
No. If you use 'covering', that word becomes important. We don't normally use 'cover' when we simply mean 'wear'. Perhaps this person has scars that he is sensitive about, and is hiding them by wearing a vest. I don't know; you used the verb.