SoundCloud the webapp already seems to have its UI designed for music playback (it's a very slick and responsive design, by the way).

Whether the UI of the standalone app is more polished that's debatable, but as far as "isolating a service to a program" and "being able to close the browser fully without closing down music" go - in and of themselves - using another browser in paralel looks sort of equivalent to it.

I don't rule out there could be some other advantages of this solution, like smaller memory footprint etc., I addressed the specific reasons given by the parent commenter

Depending on how your OS displays tabbed browser content it can be harder to access one tab/window of many in a single application than a single unique program. It also means that it has the same volume level as the rest of your browser using system volume (if anyone but me does..)

In general I'm a user who never downloads apps on my phone (unless there's something I find unsuited to my browser) - but for a music player I want it to be quickly isolated for skipping, pause, etc etc.

That's also another factor, in-browser stability is subject to each individual end user. A native client helps improve stability in the overall browser.