Mine is a pronoun that is meant to take the place of another word (much like he, she, or it) with the added meaning that it belongs to you. When using the phrase "my own," it is usually used as an adjective followed by what you're talking about (ie. I helped out my boss on my own time).

There is not much difference at all. Just two different ways of saying something is your possession. For example: "It is mine" "It is my own" The former is more likely to be said in casual discussion.