Anniversary Edition/Patches
The AE provides the latest patches needed to run Oni on modern machines.
Windows
Oni.exe
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.
Daodan DLL
The DLL, actually called binkw32.dll, comes along with Oni.exe and hooks into it when Oni launches, patching it to fix bugs and add features. See the Daodan DLL page for a list of the changes made to Oni.
Mac
Oni.app
The Mac side of things is a bit more complicated, since there have been three major builds of the game over time. See History of Mac Oni if you want to know the whole story, but all that a modern player needs to know is that the Intel build has a few differences from the original Oni, listed HERE.