You have a website http://master.com and want to have a subsite http://slave.com or http://slave.master.com which is same as main site but has another homepage and another logo displayed via a module.

Subsite should be created as parked domain for master site in your hosting panel first.

Both master site and subsite will use same database tables and hence have same contents.

You create menu items and a module for slave site in master site as usual.

MightySites allows you to hide selected modules and menu items for each site: you configure master site to not display these slave's menu items and module in master site, and hide master site-specific menu items and logo module in slave site.

It's also possible to select menu items and modules which should be exclusively displayed on this site.

Now each site displays own menu items and single own logo module, all other contents are same.

All contents are managed from backend of master site as usual.

Installing/updating components and Joomla update are performed in master site only.

Each site can use own template style and load own CSS file.

All categories and all articles are same in all sites, but note that you can control display of articles for each site via menu item settings: i.e. master site can display articles from 'Master News' only and slave site from 'Slave News' only.

Note that search will display all items, custom search plugin is required to apply your logic of content separation.