(1)I can do that on my own.
(2)I can do that by myself.

Is there any difference?

1 2
Not as far as I know.
"By myself" means without any outside help. "on my own" may mean that too, but can also emphasize the solitude.
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I live by myself.

I live on my own.

I think the first one means that I live alone; I don't have any roomates. But the second example suggests that I'm independent; I earn my living. I might have a roomate.

What do you think?
You are right on Pastel. You hit the nail on the head! (This means you got the exact meaning, in case you didn't know.)
Thanks, Danyoo. Happy New Year to you.
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No I disagree. I live on my own would also mean that no-one else lives with you, it does not mean independence from parents but with other people.

'I don't live at home' is a rather odd way of expressing the independence from parents thing (used by young people).
Can you think of an example that 'on my own' and 'by myself' are not interchangeable?

I live by myself: This means I live alone. I don't have a roommate or any other family members living with me. The meaning is very clear.

I live on my own: I found the following definitions from Cambridge.
(all) on your own
1 alone: I like living on my own.
2 without any help: I did my buttons up all on my own, Mummy.
stand on your own (two) feet INFORMAL
to be able to provide all of the things you need for living without help from anyone else:
She'll have to get a job and learn to stand on her own two feet sooner or later.
On the other hand, dictionary.com has the following meaning:

on (one's) own
  1. By one's own efforts: She got the job on her own.
  2. Responsible for oneself; independent of outside help or control: He is now out of college and on his own.
It could be the difference between BrE and AmE.
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