And that might even matter for some audiences. It doesn't for all of them: BCC had 2x the conversion for installations on Mac (where it is, ahem, distinctly non-native) relative to Windows.

Whether it matters depends a lot on how competitive the market is. If your app is head and shoulders above the competition in functionality, users will put up with a little UI ugliness. If most of the apps in your market basically do everything the user wants, they're going to decide based on what is prettiest/most responsive, which is usually the native app.

I still remember discovering uTorrent. I'd tried about 4-5 Bittorrent apps before then, including 3 wxPython-based ones and Azureus. They also basically did what I needed - they let me download, they let me keep track of existing transfers, and they let me throttle bandwidth. The difference was that uTorrent didn't take up a few hundred megs of RAM and didn't peg my CPU. The developer accomplished this by doing an end-run around all the layers of abstraction and keeping an eye on performance; it's much harder to do this with a cross-platform library in the way.

u right patio11, bendmorris should really tell us who the target audience is.

Well, if Java means Swing, you are extrapolating from a very bad example. For instance, the Qt library looks pretty native on most platforms, and with some platform-specific code (e.g. proper support for application menus for OS X) Qt programs can also feel native.

I wrote some small Qt GUI applications (mostly frontends to scientific tools), and most people don't notice it's not native.

Also, this depends a bit on your market. Some markets are more sensitive to nativeness than others, for instance, by operating system:

OS X > Windows > Linux/UNIX

In OS X people are quite used to polished native applications that conform to Apple's HIG, they will also expect it. Windows is somewhere in the middle, there are some widely used applications that do not look native. Linux/UNIX users seem to be completely oblivious to nativeness, since they are used to having a mix of Qt/Gtk+/Tk/Lesstif/whatever applications.