If you want to interact with local hardware/storage media, native apps are still easier. "Offline" mode can also be the default, at least for systems that have a local data store. Finally, multicore CPUs may be theoretically exploited with web workers, but it's a lot easier to just spin up a native thread in your Java code.
Downsides are all the usual ones, of course: updates don't happen automatically, you may need some sort of licensing/DRM if you want people to pay for your application, and your testing story is a lot more complex. (Plus: look-and-feel on various platforms, reverse engineering is that much easier, etc., etc.)