If we are using GitHub Enterprise inside the company and the Private Mode is enabled, what's different between a public and private repo?
Typically, your Github Enterprise instances are visible to developers within your organization. However, also within your organization, you might want to have a more fine-grained control of who is allowed to see which repository.
Because of that, Github Enterprise knows two visibilities of repositories - private and public.
It's straight forward, as the documentation tells us:
Public repositories are a great choice for getting started! They're visible to any user on your GitHub Enterprise instance, so you can benefit from a collaborative community.
Private repositories require a little more setup. They're only available to you, the repository owner, as well as any collaborators you choose to share with.
Edit for clarification: The Github Enterprise private mode, does regulate access on a different level: it merely defines that the forge itself cannot be accessed without being logged in.