What's the difference between single key-pair public key encryption and proxy re-encryption? Proxy Re-Encryption provides the ability to transform a ciphertext encrypted under some key to the one under another key.
This seems interesting. But, something seems strange.

Suppose Alice has a key pair (sk_a, pk_a) and Bob also has a key pair (sk_b, pk_b). If Alice and Bob generates a proxy re-encryption key to transform a ciphertext under pk_a to the one under pk_b and vice versa, they can send message without revealing their secret keys.

At this point, what about the following scenario?
Bob securely transmits his public key pk_b to Alice. Then, Alice encrypts her message vy using pk_b. There is no need to transform the ciphertext.
Also, Alice can send Bob her public key pk_a when she wants to receive the message from Bob.

What is the advantage of proxy re-encryption?
I think there is no potential difference between standard public key encryption scheme 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.