Well right off the bat: You don't have to install anything. You also don't need to worry about updating a thousand clients if a security problem crops up--you just update the server and you're done.
Also, in terms of "why" there's all sorts of neat things you can do with a web browser that aren't so easy with regular terminals. For example, displaying inline images (if you code it right). Gate One can display images, PDFs, and play back sound files right there in your terminal if you do something as simple as 'cat somefile.png' or 'cat somefile.ogg'.
Source to Gate One (also has some screenshots of the aforementioned features): https://github.com/liftoff/GateOne
The other nice thing is that you can use it through basically any proxy server (provided it doesn't "speed bump" you--breaking WebSocket connections).