Here I am replying to my own question, but I now have what I think is the correct information so I will post it in case anyone else needs to know.
1) The COTM (Conquest of the Month) is played on Civ3 Conquests. Civ3 Conquests is the version of the game that you get with the Steam download of Civ3. If the COTM is downloaded and saved to the Conquest save folder it will open, play, and save properly.
2)The GOTM (Game of the Month) is played on PTW (Play the World). The Steam version of Civ3 is C3C (Civ3 Complete) and includes a playable version of Civ3 Conquests, but NOT Civ3 Play the World. If the GOTM is downloaded and saved to the Conquests save folder it will open and can be played, but it cannot be saved and uploaded to the GOTM website for submission to the contest. To play the GOTM I needed to buy the Play the World discs and the original Civ3 discs on eBay, load them on my computer, and play PTW outside of Steam. The downloaded GOTM game must be save to the separate PTW save folder, not the Steam save folder.