Did you install Civ3 to the default location, and if so, do you have UAC enabled? That can potentially cause programs to not be able to write to their own files (in this case, the high scores file). I had the same issue back in 2007 when Vista was new, and solved it by finishing the game on my old Windows 98 desktop. But knowing more about how Vista (and later) works now, I think disabling UAC/installing to a different location would have also solved the problem.