Our organization uses MediaWiki, and I cannot see any difference when I edit a page and use the PUBLISH button at the top or the SAVE button at the bottom. Curiously, I also cannot seem to find any information about this on the MediaWiki wiki! Is it just too obvious to spell out?
Topic on Project:Support desk
Publish is added by the WikiEditor extension. It is exactly the same as Save, except it opens a dialog to enter the summary rather than using the one already on the page. It's really a non-feature which you should disable globally by setting $wgDefaultUserOptions['wikieditor-publish'] = 0; in LocalSettings.php.