To be fair, very little of what's proposed in this project changes any of those points. You can already store data locally using localStorage APIs. You can already cache application content/scripts/resources locally using HTML Manifest files, which also let you use the application when you aren't connected to the net. And of course, you can always access remote resources when you are (even across domains using the new Cross Domain XHR technology).
The crux of what they are proposing is a standard way to actually say "I want this web site to be considered an 'app' that can be installed using whatever your platform wants to consider installing". In other words, a slightly more enhanced, standard version of adding an app to your homescreen on the iPhone.
Local storage is currently limited. On the desktop there will be means for unlimited storage. Manifest files can present more than merely what file to download (such as operating in the background). By setting the apps to a different domain, it gives them room to allow for a different permission set.
If it would become unlimited than nothing would stop my web app from filling your hard drive up with crap. So removing this limit maybe not such a good idea. Besides if Mozilla wanted to provide unlimited storage they could just remove the limit from the HTML5 storage, but they won't for the reason I already wrote.
The crux of what they are proposing is a standard way to actually say "I want this web site to be considered an 'app' that can be installed using whatever your platform wants to consider installing". In other words, a slightly more enhanced, standard version of adding an app to your homescreen on the iPhone.