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 suggested however that there is no such thing as public speaking.
What they meant was that the same dynamics apply in either case and that the best public speakers use the same techniques that we all use when speaking one to one. Things like relating to the listener, behaving as the listeners expect, using a full frontal stance and many more.