v.1.22 is the last official patch of the conquests.exe (31 Civs, and new mechanics such as autoproduction and unit-enslavement, new victory conditions such as victory-points, regicide and elimination -- and a few never-fixed bugs). That is the version which is installed with Civ3Complete, so this version of the game is already fully patched. The .exe file will be installed in the ../Civilization3Complete/Conquests/ folder, and if you let an installer-package/wizard do everything for you, any 'Civ3Complete' shortcut that it created will be linked to this file (so running via a shortcut will always start up Conquests/Complete v.1.22).
v.1.29 refers to the last patch of the original 'vanilla' civilization3.exe (16 Civs, no frills -- but no bugs either). It is not intended to replace the conquests.exe. Wherever you obtained a copy of this file, it should be placed in the root ../Civilization3Complete/ folder. If you added this file to your installation manually, you will either need to run it directly from that location, or create a separate shortcut for it (right-click on the file-icon, and choose the appropriate option).
You will be able to see immediately if you have successfully run the vanilla .exe file, because the intro-movie is slightly shorter, and the background picture on the 'New Game' screen ("Tower of Babel") is different to those for Conquests ("Samurai") and Complete ("Locked Safe").
Bear in mind also that if you have installed Civ3Complete in the protected C:/ProgramFiles/ directory, Windows may have quietly prevented you from manually adding the vanilla v.1.29 .exe file to that location: it may have been VirtualStored it instead.