The primary difference between the two dishes is there is minced lamb in shepherds' pie and minced beef in cottage pie. Both dishes came to be because people would use Sunday roast leftovers to make these dishes to avoid waste.
Many people use cottage pie and shepherds' pie interchangeably. In fact, some people use beef for both or lamb for both. There are many variations, including a vegetarian variation that uses tofu in place of lamb or beef. People make these dishes all around the world, but they remain the most popular throughout Europe where people are mindful of the right meats.