"Private" domain registrations result in your domain being registered to the privacy service and not to you. At least, that was true for the ones I looked into.

That may not matter to some people, but it could be a problem to have that critical resource outside your organization's control.

EDIT: See the posts re Gandhi, whose privacy service apparently leaves the domain registered to you. https://news.ycombinator.com/item?id=7934145