I did research for a Blockchain class a few summers ago. One of the use cases for private Blockchain is if multiple competitors want to cooperate to provide a single service. The most successful example is Maersk's Tradelens. Multiple competitors can hand off a package between each other and a customer or customs official can see the whole journey of that package and sensitive data like invoices without all competitors having access to that data or having to trust a central authority to keep everyone's secrets.