Serious question, why do we need apps that work offline? Finding it hard to imagine what you can do on a computer theses days without an internet connection, besides maybe an old single player game devoid of any patches.
It's not so much that you can use it offline (though that's one benefit) but rather that it caches your assets so if there's very, very slow connection you can still use and read content. If it's something that requires the Internet to work, it can show a "no connection available" message but it still load, similar to a native app.
The goal is to avoid the "white screen of death" you get when a web app won't load.
Travel. I can still look at my ToDos, email, note, saved stories (Instapaper) on a plane thanks to offline apps. I can use it as a clock/alarm/calendar while in another country. My phone can even be used as a GPS when offline thanks to apps like GPS Nav that allow navigation using saved map data.
Always-online is a luxury of spending all your time in certain parts of the world. Baking that assumption into all apps diminishes their utility greatly.