Well, that’s a good question! The most obvious one is in the name: co-browsing is browser based, meaning that you only share the browser instead of sharing your entire screen (which is the case with screen sharing).
Screen-sharing solutions are pixel-based, the drawback of this approach is that screen updates are slow and of low quality. What most screen-sharing tools have in common, is that they require external software to be installed by the users. This makes it unsuitable for most web situations as people are often unwilling to install extra software that circumvents the browser's security measures.