Unlike screen-sharing, co-browsing can be more interactive. As the agent, you're able to draw, click, scroll, and type on the client's screen with co-browsing. It's also a lot easier for the client as it doesn't require them to download any additional software.
Another big draw with co-browsing is that it offers more privacy to the client. As the agent, you're restricted to the webpage and do not have access to the client's entire browser or desktop.