The difference between private proxies and shared proxies mainly lies on their exclusivity. Private proxies are only used by one person. Hence, you don’t have to worry about the bandwidth speed because no other person can have access to it except you. Some applications also work much better with private proxy servers.
In shared proxies, you share your IP address with several internet users. With shared proxies you run the risk of getting banned in case one of the users in the Shared Proxy gets banned. However, Shared proxies are cheaper than private proxies.