"I will do it myself" can mean two things:
I will do it without any help
I will do it also.
"I don't want any help. I will do it myself." (1st meaning)
"We've just visited the beach." "Oh, I'll do it myself in a minute." (2nd meaning)
However, "I will do this on my own" has only one meaning - to do something with no help from anyone else.
[Note that Chris Conlon makes a good point, below. You would usually hear this in the form "I'll do that myself soon." However, you will hear "I'll do it myself..." in that situation - just not as often.]