In many cases, they are interchangeable. However, there is a subtle difference. "On my own" is used more for doing something with no help, where you would usually expect help - also when making a decision when it might be expected that you would try to get somebody's permission or agreement. "I repaired the car on my own." "I decided to go ahead and start the process on my own because I couldn't wait for a decision." "By myself" usually describes a situation, not something being done. "I spent Saturday night by myself."

It has almost no difference. You could use both in almost any case. As I am no teacher I checked what others say. Here is what I think you should take in consideration: "I'm studying English by myself" means to me that you sit alone and study English. "I'm studying English on my own" means to me that you have taken on the responsibility of studying English with no help from anyone else. As part of that study you might have taken on the task of striking up conversations with native English speakers. In other words, it doesn't mean that you will always be alone when you are working on your English studies. (source: http://forum.wordreference.com/showthread.php?t=280005)