Form tabs aren't saving in the Sprout Forms plugin
OK, heard from Barrel Strength / Sprout support. His reply:


  Please create any fields that you need before adding the tab, then
  please go ahead and add the tab(s) then drag and drop the fields that
  you need into the new tab (rename the new tab in this step if you
  want) and finally save the form. The new tabs need to have at least
  one field to be saved with the form.


Since you can't create an empty tab, the trick is to create all your fields first -- or at least, one field per tab. Then, after you create the tabs, you can drag a single field to each tab and save the form. That will keep the tabs. After that, when you create a new field, you can select the tab it belongs to in the dropdown.

I think that's how it also works in the native Craft field manager? So hopefully this helps somebody even if you're not using Sprout Forms (which is awesome, by the way.)