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."