What is the precise difference between a private app and a public app? If anyone has either of the URLs they can access it. In addition to the fact that a public app has a human-readable URL name?
Is the URL to a public app available in some directory on Anvil, which makes is less private?
The difference is simply the URL. The private URL has sufficient entropy that nobody can find your app by accident/guessing. You can also re-generate the private URL if you want to invalidate links you have previously given out (with a public URL you would have to think of a new human-readable URL!)
There is no public directory of Anvil apps (but if you want to share your app with the community, you can post about it on the Show and Tell section of the Forum, as well as tweeting about it and all the other usual ways.)