One possible reason for Firefox to work a bit differently in a private window -- besides using temporary storage for cookies and cached files -- is that some of your add-ons may not be allowed to run in private windows. You can check on that as follows:

Open the Add-ons page using either:

• Ctrl+Shift+a (Mac: Command+Shift+a)
• "3-bar" menu button (or Tools menu) > Add-ons
• type or paste about:addons in the address bar and press Enter/Return

In the left column of the Add-ons page, click Extensions. On the right side, find the "Manage Your Extensions" heading.

Extensions which are allowed to touch tabs in private windows will have the purple mask icon. The ones without that icon only run in regular windows. Are there any regular-window-only add-ons that could be blocking bits and pieces of Google data or scripts?

Another user said it's caused by Adblocker Ultimate (here). If you have that add-on, see whether you can update it since hopefully they will quickly issue a fix.

......

Google Drive won't load with Firefox unless in Private mode.

Probably means that an add-on is causing the issue. It would be helpful if you show us where it shows your offline. I have no issues with GD. see screenshot What OS? What version of Firefox?

Happy Thanksgiving!

I submitted a support request to the company and got this message back overnight:

From: Alex from AdBlocker Ultimate <support@adblockultimate.net> Hello Please update the adblock filters in the browser extension settings. The issue will be fixed.

Does that work for you?