The first means "by oneself" in the sense of not relying on someone else to accomplish something, whereas the second means "alone" in the sense of doing it as one person. For example, you can say either 自分で学校に行きました／一人で学校に行きました, but the former almost implies this sense of "I did it all by myself! I didn't need any one to take me," whereas the latter sentence just means you went alone.