I bought Rainbow Six 3 on Steam and it includes two games in my library, "Rainbow Six 3: Gold Edition" and "Rainbow Six 3: Athena Sword". What's the difference between these two, and which should I play?
The game you purchased on Steam was "Rainbow Six 3: Gold Edition". This includes Rainbow Six 3: Raven Shield, and its first expansion pack, Athena Sword.
As Athena Sword is an expansion pack, and the story appears to continue from the story line in Raven Shield, you should probably play through and complete the library entry in Steam called "Rainbow Six 3: Gold Edition", and then continue the story as originally intended by playing "Rainbow Six 3: Athena Sword".
For more detail on what the Rainbow Six 3: Gold package contains, see the Steam page.