What are the differences between Private and Public Cloud Computing?

A.
Private cloud is a solution you own where Public is usually a hosted, ISV based, or purchased externally cloud system that is normall a Pay-as-you-Go service.

Explanation:
To understand the difference I found this helpful and we should all read it. I made the answer up and is the biggest difference, in my opinion.

http://blogs.vmware.com/vcloud/2011/04/enterprise-private-cloud-public-cloud-the-chaos-monkey.html