There's nothing wrong with "write it down on a scrap of paper" in AmE, but it's not exactly the same as "a piece of paper." If you have a nice new sheet of paper in good condition, you would call it a "piece of paper." If you tear a corner off of a newspaper or an old envelope to write something down on, you would call it "a scrap of paper."
"Scraps" basically means "small torn or leftover bits." You might give your dog "table scraps" -- the odd bits left on people's plates after dinner. If your car has been in a terrible accident and can't be fixed to be drivable, you might be able to "sell it for scrap" -- that is, sell it for a small amount of money to someone who will just take it apart and use whatever bits can be salvaged f