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.

Choosing between Private and Public IPs for Virtual Servers has other ramifications as well. Example: If your are using the GTM module, using Private IP addressess and specifying translation on your vips will hinder Virtual Discovery from working:

sol9138: The BIG-IP GTM system disables virtual server auto-discovery for BIG-IP systems that use translated virtual server addresses http://support.f5.com/kb/en-us/solutions/public/9000/100/sol9138.html?sr=35087534