Why does Bank Reconciliation say there is a Difference, when there is not?

That behavior is indeed strange, The Old Oracle.  

There are times that your browser becomes full of site data causing problems in QuickBooks Online. To check if this is a browser-related issue, you can open your QuickBooks using an incognito or private window. This can help us rule out any cache-related issue that might've caused the problems. Use these shortcut keys to open it: 

• Google Chrome: Press CTRL + Shift + N.
• Safari: Press Command + Shift + N.
• Mozilla Firefox: Press CTRL + Shift + P.

If it's now working well, you'll have to go back to your regular browser and clear its cache. This will delete the accumulated junk files and enhanced your browsing experience.  

When you're done clearing the cache, close any browser instances to complete the process.  

Also, make sure you're using a supported and updated browser for QuickBooks Online. 

You can also utilize this article to reconcile your bank statement in QuickBooks Online: Fix issues the first time you reconcile an account in QuickBooks Online. 

I'm also up and willing to help out again if you have questions regarding your QuickBooks. If you have any other tasks in QuickBooks, please let me know, and I'll extend my assistance to those areas.