What's the difference between single key-pair public key encryption and proxy re-encryption?
With two parties, using proxy re-encryption does indeed not make much sense. It becomes interesting when there are more parties. For example, suppose you receive encrypted mails, that transit through a server. Then you could want to forward one of your encrypted mails, stored on the server, to someone else (say, Bob). With a proxy scheme, you can just ask the server to convert this encrypted mail into a mail encrypted with the key of Bob, and to send it to Bob. In general, it provides a convenient way of managing databases of encrypted files in a multi-user scenario.

You can find more applications in the introduction of any paper on proxy re-encryption. For example, page 3 of this paper discusses examples in more details. This paper also mentions other applications (with references) in the third paragraph of its introduction.