Well, I searched and searched. I did strace's on both machines and multiple different output comparisons. I tried. I really, really tried. But after several hours I have to ask for help.
Application running on system #1, which is Ubuntu 10.04 LTS, Gnome2, running Wine 1.2

Notice the borders, the centered lines surrounding "APPLICATION SELECTION" (at the top), etc.?
Now the same app on system #2, which is Linux Mint 13 LTS, Mate 1.4, running Wine 1.4

Ugly. Ugly and annoying. Very annoying, in fact. (Mostly because it was good on the old system and is b0rk3d on the new.)
The one striking thing I noticed, in comparing strace output, is the former system opens and stats the ".fon" files in /usr/share/wine/fonts, whereas the new system does not. (I tried copying all those into my local Wine Fonts directory. Didn't help.)
That isn't the only difference. There's also /usr/share/fonts/truetype/freefont (there, but not touched) and a few others.
Help?
Thanks,
Jim