"I will do it myself" can mean two things: I will do it without any help OR 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.]

"I will do it on my own / I will do it myself": Honestly, I see no difference between these two sentences except the wording. The meaning is the same. They are just two ways to say the same thing, as in this example: "I want to work tomorrow / I would like to work tomorrow."

Imo they are both very similar are probably interchangeable in many sentences. If were to pick a difference, it would be that, "I will do it on my own" often implies that the person will do the task whilst alone.