Public vs Private Virtual IP

The only significant difference would be "routability". The generally established private IP subnets are defined here:

https://tools.ietf.org/html/rfc1918

as:

 10.0.0.0/8
 172.16.0.0/12  
 192.168.0.0/16    

and are basically ranges of addresses that are considered private and will not route across the Internet. They can route internally in your environment though. So I suppose to more specifically answer your question, a private address is any address in one of the above ranges that you might apply to a VIP that either is not directly connected to the Internet and/or is used for internal testing/development. An external IP address is anything not in the above ranges and would be directly accessible across the Internet. There is otherwise no functional difference between the two as far as a virtual server is concerned.