The game has a lot of problems on vista, i had the same troubles.
If you choose "Run as administrator" then that may clear things up. However, the game may still lag horribly from Vista acting up. The best thing to do is get eduke32, which makes DN3D run really well without dosbox or anything causing lag.
Eduke32 can be found here: http://www.eduke32.com/
Download it and be sure to place the eduke32.exe file in the correct Duke 3d folder. Just follow the instructions and run the game using eduke32.exe, it should work really well.
Hope this helps!

Let me guess, you're logged into Vista under a standard account, right?
Duke has to write files to the program files folder and Vista usually doesn't allow that which is why right clicking on the Duke3d shortcut and clicking "Run as administrator" works. Administrator accounts allow write access to those folders. If you don't want to do that every time you run Duke do this instead:
right click on the program files/gog.com folder ==> go to security. Click "edit". Under "users" click the "full control" check box then click OK.
Duke should run normally when you click on the shortcut now.