No, it does not; it's the same content / same game.
The web download (ChessCompStompWithHacks_Web.zip) is just an html file; you can double-click it to open the game in a web browser. You can use this if you want to keep a local copy (which works without an internet connection).
The Windows (ChessCompStompWithHacks_Windows.zip) and Linux (ChessCompStompWithHacks_Linux.zip) versions are desktop versions of the game. It's the same game, just in a native format (e.g. on Windows, it's an .exe file that runs outside a web browser).