Generally public speaking is more formal and more clearly articulating. In public speaking, you're usually trying to persuade a group of people to act or think a certain way, so you want to present...
Public speaking generally means speaking in front of an audience of more than one. Private speaking probably means a one to one conversation meant fro no-one else's ears.
One book I read recently...
Pretend like your sick, don't go to school, pretend like you can't talk, or you could kill your teacher(JUST KIDDING ON THAT LAST ONE!!!!) it's totally easy to get out of publically speaking in class.