The main reason that some IE websites don't work with Firefox is mainly due to the Active X controls. Active X is a technology from Microsoft to add functionality to Windows programs. Active X, basically, is a proprietary object of Microsoft; that is, Active X is designed to work only on Microsoft Windows operating systems.

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Because of this, Firefox does not utilize Active X, as it restricts websites and operating systems to just that of Windows. This prevents users of Mac OSX or Linux systems from viewing certain websites. There is also the security issue, as Active X has been a primary target of malicious software by hackers, spyware, and viruses.
Firefox uses plugins and extensions in order to add functionality to its browser. These include plugins for Flash, Adobe Reader, browser themes, and a variety of different user created applications.