Task 'SharePoint' reported error (0x80004005): 'An error occurred either in Outlook or SharePoint. Contact the SharePoint site administrator.'
After some additional troubleshooting, I determined that this only happened when trying to synchronize a sub-folder of the library (with the amount of data in this library, synchronizing sub-folders was the common practice). Eventually a call to Microsoft PSS was required to troubleshoot this further.
In the end, we determined that parts of the folder structure used characters that Outlook considered invalid for synchronization. Specifically, Outlook would not sync a folder with "--" in the name or folder path (in our case, the root folders were all named "A---", "B---", etc.) The support engineer did mention that there may be other characters which Outlook won't sychronize, but luckily we did not run into those.
If you are running in to this Outlook error when connecting to a SharePoint document library, try looking in to the folder names and see if something looks abnormal.