The Anniversary Edition provides the patches needed to run Oni on modern machines, as well as various improvements. The following two files, in Windows, or one application bundle, on Macs, are installed automatically with the AE.
The Oni.exe provided by the AE is more or less unmodified since Oni's release in 2001. It is only included so that the Daodan DLL has a consistent application to work with, since the player's retail copy of Oni might have a slightly different EXE due to localization, etc.
This DLL, actually called binkw32.dll, sits alongside Oni.exe and hooks into it when Oni launches, patching it at runtime to fix bugs, add features, and unlock Developer Mode. See the Daodan DLL page for a list of the changes made to Oni.
The game application currently installed on Macs by the AE is a modern Intel build of the game, with a few differences from the original Oni that are listed HERE. It contains higher-quality intro/outro movies ported from the PS2 version of the game, which Oni places in the GameDataFolder when it first runs.