It's surely a quite straightforward distinction:

Public transport = bus or train that anyone can get on, private transport = car for personal use

Public company = one with tradeable shares that anyone can buy, private company = one with non-quoted restricted share ownership.

So I image public plane = scheduled air flight, private plane = one just for you.

I could go on:

Public boat (more likely ferry, or hire boat) = one available to anyone , private boat = just you and me.

Public house = pub where anyone can drink, private house = mine.