mkdir: /Users/root/Library/Caches: No such file or directory
sh: line 10: /Users/root/Library/Caches/com.apple.SoftwareUpdate/locale: No such file or directory (1)

Now I get that AppleScript error when I enter the password for the AMD Software Update.

You must have user password set to run ASU. It's bug(?) which I think will never be fixed, as unfortunately ASU source files are lost. So, go to: System Preferences > Accounts. Select your account on the list, click on "Change Password" button. Enter password, click another "Change Password" button to close dialog. Exit System Preferences, run ASU. Enter you new pass, it should work now.

Or forget ASU and download Zephyroth's AMD Patcher. Download packages using regular Software Update, then patch it using this little util.