Gyros, the Greek variety is a sliced loaf containing a mix of Lamb and Beef, and is about the toppings served on it.
Shawarma can be Chicken, Goat, Veal or Lamb and is more about the flavor of the meat.
Gyros is often also sold as Shawarma in western countries due to their similarities, but if you want real Shawarma go to a place where Arabs eat and discover the difference. I hope this helps. Cheers!