A public help center is a portal where your users can find the information they need without necessarily getting to your customer support team. It allows them to learn more about you, your products and services as well as self-educate on how to use them without much hassle. This site, for instance, is the help center of helpcenter.io (thus the name self.helpcenter.io :)).
Here's one more example you may have seen, it's the Help Center of Pinterest:
On the other hand, a private help center might be a place for you team to keep and organize all the information about things that need to be shared so they can more informed thus more efficient.
If you look around, almost every business has a help center. Facebook has one. Twitter also has one. And Google as well. GitHub too has a help center. I'm stopping here as I'm pretty sure you don't need more examples to see that having a help center site that allows your users to easily find information is crucial to your customer success efforts. And the fact that every major organization has one, proves it.