It just shows an hourglass for a second then stops.
Have you tried rebooting? Upgrade to Microsoft Vista? Upgrade to OS X?
What this problem usually means is that the application can't find some resource file in Microsoft Windows, and so it quits. This is the same problem that prevents some applications from running, with the claim that they "need" Microsoft 2000 or XP, but once the version check is disabled, the application runs normally.
Something probably damaged your iTunes configuration such that it can't find the files that it needs to run, or maybe your copy of Microsoft Windows is really missing some files.
An obvious thing to check is to make sure that "double-clicking on iTunes" really means that this user has verified that the iTunes icon is pointing to the correct executable file, that the executable permission has been set, that the user account has the correct permissions to read those files for iTunes, and that there are no other programs that could be interfering with Microsoft Windows. Check for offending applications from Symantec, McAfee, and other "security" companies that like to cause problems for Windows.