I'll add to this and this may be my own opinion but it's valid I think...
Firstly we haven't changed the name or tooltips for the buttons because we try to avoid changes that make existing guidance outdated unless there's good reason.
Also technically what Fay describes is accurate in that Play/Preview opens the project to you as the author or co-author and saves and displays your changes via preview.xml whereas Publish doesn't play the project but does Publish/Save your changes to both preview.xml and data.xml which is used when viewing the public link.
Historically this was useful, as Fay describes, because you could make changes without yet making those changes available to the public version. However I would argue that is the old way to keep changes from showing because we now have the ability to hide pages and sections until we wish them to appear and either manually or via date/time options.
So personally I click publish regularly and habitually and regard it as Publish/Save my changes to both xml files. If my project is public and already used by learners etc then I can make new hidden content and unhide when ready.
Finally always click publish before exporting otherwise you exports may not contain all your changes.
HTH
Ron