If you say "My hair began to thin" it implies that your head is still covered with hair, but there is less of it; you no longer have a lush, full head of hair, but hair is still in the same places it used to be. If you say "I started to go bald" it means that on parts of your head (usually the top) you no longer have any hair, but you still have hair in other parts (such as the sides or the back).