Not all the world has good Internet.
If all you have is a 3kbps connection tethered from a phone then it's good to SSH into a server and browse the web through, say, elinks. That way the server downloads the web pages and uses the limited bandwidth of an SSH connection to display the result. But traditional text-based browsers lack JS support and all that other modern HTML5 goodness. Browsh is different in that it's backed by a real browser, namely headless Firefox, and uses that to create purely text-based version of web pages and web apps that can be easily rendered in a terminal or indeed, somewhat ironically, in another browser. Though note that currently the browser client doesn't have feature parity with the terminal client.... More Info »
Why not VNC? Well VNC is certainly one solution but it doesn't quite have the same ability to deal with extremely bad Internet. Also, terminal Browsh can use MoSH to further reduce bandwidth and increase stability of the connection. Mosh offers features like automatic reconnection of dropped or roamed connections and diff-only screen updates. Furthermore, other than SSH or MoSH, terminal Browsh doesn't require a client like VNC.
One final reason to use terminal Browsh could be to offload the battery-drain of a modern browser from your laptop or low-powered device like a Raspberry Pi. If you're a CLI-native, then you could potentially get a few more hours life if your CPU-hungry browser is running somewhere else on mains electricity.