Weathered things are worn by use, time, or weather, like the weathered shingles on a beachside cottage or an elderly man's weathered face.
Your favorite sneakers will look weathered by the end of the summer. And, the weathered Canadian flag your brother hung outside your house back when he was a hockey fan should probably be replaced with a Union Jack now that his favorite spectator sport is cricket. Weathered comes from the verb weather, which first meant just "come through safely," before gaining the meaning "wear away by exposure," like the paint on a boat over time.