Do you use Akeeba to clone the sites?
This will still copy the html and the database, but gives you options to use a different prefix for the seconds database and some other options.
Might make life easier if your not already using it.
I dont know of anything that will run 2 sites off 1 database as you would be practically rebuilding Joomla! to do it.
I cheated with a site a while back where i used 'Flexi Custom Code' plugin (which allows php/js/html) and then used some php script to display relevant info depending on what subdomain was used.
So...
Install Flexi Custom code and use something like
<?php
$url = $_SERVER['HTTP_HOST'];
if ($url == 'www'){ ?>
<p>THIS IS SOME NORMAL TEXT FOR PEOPLE WHO VISIT VIA http://www.MYDOMAIN.com</p>
<?php } else if ($url == 'help') { ?>
<p>THIS IS SOME TEXT FOR PEOPLE WHO VISIT VIA help.MYDOMAIN.com</p>
<?php } else if ($url == 'contact') { ?>
<p>THIS IS SOME TEXT FOR PEOPLE WHO VISIT VIA contact.MYDOMAIN.com</p>
<?php } ?>
Obviously this would work to show certain info to certain people, you can also include modules into this too using the {loadposition xxxxxxxxx} command.
This wouldnt work for showing/hiding specific menu items though, unless you wrote some more manu stuff with php/flexi.
Hope this helps a little, but i feel you may be wanting more than just a couple of pages different?