The NETWORK section tells the browser which resources are only available in online mode. An asterisk indicates that anything that isn't cached is a network resource.
Application Cache is a handy tool with almost no downsides if used correctly. If you simply cache everything on your site, you (and your site visitors) will soon be wondering why the content never changes. If you cache only resources that don't change very often, or are diligent about keeping your cache manifest up to date and releasing a new version when you update files, then Application Cache will make your application work offline as well as be much more responsive in online mode.
Local browser storage has undergone a major upgrade in the last few years. The variety and similar names of the different implemented APIs and recommendations have led to quite a bit of confusion over what's OK to use now versus what should be left alone. The bottom line is that different ways of storing data in the browser are available — and each has its own place.
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