On my own, by myself

On my own and by myself both mean "alone." The important thing to notice here is the preposition for each one. Both expressions are often used when we're talking about being independent. The second one comes up a lot with children, for example, "Today Nicholas tied his shoes by himself." This means that he did it without any help. Sometimes we add all to give more emphasis, for example, "Jenny made this cake all by herself!"
On my own is often used with living, to talk about living alone after leaving the family home, or after living with other people or with a partner. For example, "After his divorce, Jim decided to live on his own for a while."
The image here is by Buson, and it's called Landscape with a Solitary Traveler. We could say that the person is traveling by himself. Some people really enjoy traveling on their own, but most seem to prefer to travel with others. What about you?