"I'm on my own" can be a positive thing. A student, for example, who gets his first job and can afford his own apartment rather than live at home might say, "Things are great! I'm on my own now." It means that he is supporting himself and not depending on anyone. I'm all by myself" usually expresses loneliness. It can, however, simply emphasize the fact that no one else is around, without any emotional connotation. A person in a small office might respond to a phone request to run out and pick something up by saying, "I can't leave the office right now. I'm all by myself", meaning no else is there to handle business if he or she leaves.

I think that you would have to ask about 'on my own', rather that just 'my own'. Apart from that, they are interchangeable. I want to be on my own = I want to be by myself. I can do it on my own, thanks = I can do it myself, thanks.