Those are called "contractions", and they have the exact same meaning. So yes, "do not" and "don't" both mean the same thing. (In the case of contractions, the apostrophe stands in for the missing letters.) Contractions work well in informal speaking and writing. For formal writing (academic writing in particular) you might want to avoid them.

They mean exactly the same thing, so it is your personal preference to use either when you speak.