BitTorrent Surf is a full-blown BitTorrent client packaged as a Chrome extension
BitTorrent, the company, has been working on both educating the public on the benefits of the technology and expanding its use. BitTorrent Surf, a BitTorrent client that also happens to be a Chrome extension does the latter. It makes it possible to put a full client inside your browser and grab content via BitTorrent without having to install anything.There are plenty of benefits to this, for one it's easier to install an extension than a full-blown client like uTorrent.
Some people may not have the experience to install a desktop app, but they'll be able to install the extension.
In plenty of cases, people aren't allowed to install apps, but they can grab the extension. It's also a great addition to Chrome OS users who have to rely on web apps exclusively.
There are other small benefits to using the Chrome client, the integrated search box should greatly improve discovery, you simply search for whatever you need, click on any of the results and get the content without any hassle.
Also impressive is a complex application like a BitTorrent client, which handles dozens of connections to peers around the world via non-standard web protocol (i.e. not HTTP), built entirely with web technologies. It's a testament to what the web can do these days.
There are a handful of apps that still can't be "done" using web technologies, but the list gets smaller and smaller. There will always be a need for powerful native applications, but the day when regular users will be able to do everything they need on the web is fast approaching.
That said, BitTorrent Surf is probably not the best example of a pure web app if only because it's not.
It's a Native Client app, a technology only supported in Chrome that enables apps written in C/C++ to run inside Chrome. It also uses a lot of Python code, while Python can be considered a web technology, it's a server-side one.