properly has to do with the way you perform an action, for example : "close the door properly so the dog can't get out". Appropriately has to do with the cultural or social norms so that you do not offend anyone or get in trouble. for example : "we are going to an expedition in the desert, dress appropriately"

Properly - exactly, correctly, satisfactorily The task must be done properly. You need to properly follow the recipe, otherwise the dish will be awfull! Appropriately - in a way that is suitable in the given circumstances. He behaved appropriately during the party, everybody was happy. You cannot wear those pants to a wedding, you need to be dressed appropriately!