The two sites are probably using a different version of PHP. The one with the explanatory message uses PHP 7.0 or 7.1 whereas the other one is on PHP 5.x.
I'm not an oracle, I know how PHP and Joomla! work :) In PHP 5.x the missing class is a PHP fatal error which results in a 500 Internal Server Error as it cannot be handled by user code (like Joomla!). You'd only see an error message if Error Reporting was set to All or Development in Joomla's Global Configuration. This makes it hard for developers to troubleshoot stuff, so PHP 7 changed this behavior. Now this kind of an error is a PHP Exception. Exceptions "bubble up" all the way to application's (Joomla) error handler which will then render an error page with the exception message (i.e. the error message which gives us a clue on what's going on) and, if Debug is enabled on the site, an error trace.
So technically we're talking about the same issue, same error message, rendered differently due to different PHP versions. That's yet another reason to use PHP 7: troubleshooting is infinitely easier. You can almost always see an error message with a useful clue without having to modify configuration.php, php.ini / .user.ini and/or hunt down the server and the PHP error logs.
Nicholas K. Dionysopoulos
Lead Developer and Director
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