Depending on who the web host is application web hosting can mean a lot of things. Most of the time application hosting means that the shared web host setups your account and you can go in and use the 1-click installer to install the application you want.
Applications that can be installed via different 1-click installers include: Joomla, WordPress, Drupal, Magento, MediaWiki, and phpBB. There are more applications that many different web hosts promote as “application hosting”. The different applications vary from web host to web host.
Most shared web hosts that offer application web hosting will probably rely on either the 1-clicker installer program to maintain updates, or have their own unique updating system to make sure your application is up to date. When new updates are available your web host may auto update the application, or alert you via inside the control panel to update.
Is there anything special about application web hosting?
If the web host is tailored around to providing support to one specific application then they will offer unique features. If the web host is a general shared web host then there is probably not many unique features. When comparing shared web hosting to application hosting. If both the plans cost the same and list the same features then there is a good chance it is just marketing and not much technical difference.