It’s not a web app. It’s an app you install from the web. (blog.forecast.io)

I think the key here is: "Avoid replicating the default iOS styling and interactions".

Avoiding the "Uncanny valley" some apps can compete with native.

how did you manage to make it look one way when you viewed it in your browser, and look different once you launched the app from the homescreen. Is there a script for that?

One key advantage of installing as a web page on iOS: you get to benefit from the JIT compiler for Javascript. iOS apps that use Webkit still aren't allowed to have Javascript compiled to native object code.