My iTunes has suddenly gone off,

Hello there, brianblue.

The red iTunes icon is now the default color for iTunes 12. When getting "Error 13001" when syncing, the following Knowledge Base article provides some great steps to follow for troubleshooting:

iTunes: "Error 13001" when syncing - Apple Support

iTunes for Mac

Determine if content causes this issue

• Try to sync one type of content at a time, such as apps, music, podcasts, or photos, and determine if a specific type content causes the issue. If the syncing issue is isolated to a specific type of content such as purchased music, try syncing imported music instead. If this resolves the issue, replace the bad content with copies from a recent backup.
• If the sync issue occurs while syncing photos, try to sync from a newly created folder with undamaged photos in it. You can use your default Desktop Pictures folder that is included with Mac OS X:
  1. Copy the Desktop Pictures folder to your Desktop. This folder is located in /Library/Desktop Pictures/.
  2. In iTunes, choose to sync the copied Desktop Pictures folder to your iPhone/iPod.
• If the sync issue occurs while syncing music, podcasts, or apps from iTunes, try to sync after rebuilding the iTunes library by following steps in How to re-create your iTunes library and playlists.
• If the issue only occurs while syncing with iPhoto, try syncing after rebuilding the iPhoto library by following steps in Rebuilding the iPhoto library.
• If all types of content cause this syncing issue, test syncing in new administrative user:
  1. From the Apple () menu, choose System Preferences.
  2. From the View menu, choose Accounts.
  3. Click the "+" in the Accounts window (you may be promoted to enter your administrative password).
  4. Give the new user account a name, short name, and enter a password.
  5. Enable the option to "Allow user to administer this computer."
  6. Click Create Account.
  7. Once you've created the new user account, log out of your current user account and then log in to the account you just created.
  8. Attempt to sync your iPhone/iPod in iTunes.
• If the issue is resolved while logged in to your new user, use Activity Monitor to compare what processes are running in the older user that are not active in the new user. It may help you compare what is running in each user by sorting the processes with the My Processes filter. To quit a process, click to highlight the process and then click the Quit Process stop sign in the toolbar.

If the issue is not resolved, try syncing while connected to a different, known good computer. If the device syncs on another computer, then the issue is system specific and further troubleshooting will be required to resolve the issue. If the issue is reproduced with a known good computer, then the device has an issue, not the computer. Preserve your latest backup (see iPhone and iPod touch: About backups), andrestore the device being sure to choose "Set up as new" after restoring.

Thanks for reaching out to Apple Support Communities.

Cheers,

Pedro.