It's possible that I'm wrong but you can give this a try. Open Options/Privacy. At the top choose 'Firefox will' and set it to "Use custom settings for history"

Check 'Clear history when Firefox closes' and then click the 'Settings' button to the right of that. For those settings you can click to automatically clear whatever you want except "Browsing History." That seems to be the way it works best for me.

Also, be sure to watch how you close multiple windows. The last exited window should be the window that contains your App Tabs.

Thanks iowaman. That method does seem to work, but part of the point of clearing my private data is clearing the browsing history. I guess there isn't yet a way of making app tabs a specific exception. It would be a nice feature though.

It is best to use File > Exit if you have multiple windows open to close them all at once to avoid losing your setup if there is a forgotten pop-up window still open.

cor-el is exactly right

I use numerous "short cut" buttons from the add-on "Toolbar Buttons" - https://addons.mozilla.org/en-US/firefox/addon/toolbar-buttons/ - which has a nice red X button to Exit. I placed it on the left away from the Windows program shut-down 'X' to 'train' myself to exit properly to save my App Tabs. It's working every time now.

The issue is not a matter of multiple windows. I only have one window open. Clearing browser history clears app tabs, which I think is a pity.

It's a matter of both (or either). But I guarantee you that multiple windows closed in the wrong 'order' will wipe your App Tabs.