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The Complete Edition contains the base game, Warlords, Beyond the Sword, and Colonization on a single DVD. It contains an updated DirectX installer and contains no DRM. The individual games still require patches to the current version.
The tutorial for the game is only available through the base game installation, and the Warlords expansion contains exclusive scenarios which are only available through its installation. All other gameplay contents and features are included in Beyond The Sword; if you do not need the base game tutorial and Warlords scenarios, you may uninstall them respectively. Colonization is a standalone game which does not overlap with Beyond The Sword.
A very thorough set of instructions on how to install Civilization IV and its expansions are available here.