InfinityWiki provides a Joomla site with the tightest possible integration with a MediaWiki site, not only creating a single user sign-on, but virtually turning MediaWiki into a Joomla component.
The content management system is amazingly flexible, especialyl when equipped with a CCK, such as K2 or EasyLayouts. However, turning into in a Wiki would not only mean reinventing the wheel but also a lot of work.
InfinityWiki merges Joomla and MediaWiki into one website. Both applications continue to exist, more-or-less, unchanged, which is great for upgrades. InfinityWiki runs MediaWiki inside Joomla's main content area, provides it with a "skin" and Joomla menu modules with all the necessary MediaWiki menu items that you normally find on its sidebar.