What did they take out precisely? The voxel display? Civ III used DirectX instead. Unit design? Civ II's units were kept in a plain text file, Alpha Centauri merely had the ability to modify that file on the fly, Civ III had the ability but didn't make use of it the same way. The only changes to the engine which couldn't be done by an end user in Civ III really would be the net code, which they added with Play the World!, and as Mr Reynolds (IIRC) stated the reason it required an expansion rather than being in from the start was that the gameplay of Civilization was never designed with multiple players in mind.
The engine for Civ I - III is the same, with minor tweaks and multiplayer added in as time progressed. Civ IV is the first new engine for the series, primarily built to take multiplayer into account from day 1 rather than fudging it in later. In addition, the game now uses Python scripting rather than plain text files for storing the 'soft' rules, which adds more advanced modding capabilities both for players and the developers.
To save more dumb questions later, the engine of a game isn't the graphics (though that might be a sub engine, or in the case of a tech demo game like Crysis it might be the point of the game), the user interface or the like, it's the hardcoded rules which underlie most gameplay features. For example, one of the jobs of the Civ engine was to take the unit.txt file and convert the information there into an actual unit. The actual unit stats are 'soft' rules which could be altered quite easily without making a huge difference to how the game works; what the engine determines is the 'hard' rules which dictate what a unit actually is and the valid criteria for the information in the unit.txt file. For an example, compare how all previous Civ games (including Alpha Centauri and Colonisation) treat units compared to how they work in Civ IV - things like Civ IV's experience/unit abilities simply couldn't be implemented under the old Civ engine.
