The "maintenance menu" is doing something different to the database fix present in the extension manager.
The joomla one only work on the extensions that describe the schema and update.
The JMS "maintenance menu" is able to discover the SQL files that are used when you install an extension and after to compare the "install" SQL with the current DB structure to verify that everything is correct.
This is like that that JMS maintenance is able to detect missing update in Joomla database fix because the Joomla one does not compare with a full install but only some statement with other that may be skipped.
In the JMS "maintenance menu", when something is not understood and skipped, you also have the opportunity to apply it.
In addition, when you detect a problem somewhere, you are also able to execute any free SQL statement to apply on a website.
In addition to this "basic" functionlities, the maintenance menu is also able to create a "legacy DB" that can be used by different joomla version.
Because the DB structure are different between the different Joomla version, the legacy mode is able to compare the structure and just add fields (and not remove drop fields).
The objective of the "maintenance menu" is to be able supervise the consistency of your all websites and provide you a tool to potentially fix it on a specific site or apply a specific DB modification in case where you would find an inconsistency somewhere.
The JMS 1.3.30 is now released and you can now see the detection of schema version.
See the FAQ for the procedure to get the latest version
www.jms2win.com/en/faq/faq-joomla-multi-sites#cat-122