We can all play against the AI at any time, we don't need GoTM for that, instead it is a competition against each other.
For me an important reason to play GOTM is that it is there. It is this single map that i have a choise to play for a month and i make it the game to play.
If i do not play GOTM, civ for me usually is like this:
I start a game, the start position doesnt look good. I restart.
I restart 10 times, but the start position actually never gets better than that second or third start i got but dont have the save file anymore.
Frustrated, I play hearthstone.
I start a civ game again. This time i am structured, i start 10 games, keep al the saves and select the one i like most.
I play for a few hours, things go well, but not perfect. (they of course never go perfect because perfect is a theoretical something that just doesnt really happen)
I discover some ancient game like populous the beginning for download and play that for a few days, when i get back to civ, i dont feel "in the game" anymore.
Instead of completing the imperfect last game that i am no longer really "into", i start a new game with the intention to do it more perfect. Of course when i start a new game, i start in a desert or tundra so the process starts all over again.
GOTM solves most of that. I still abandon most games before finish, because enough went wrong to make me want to play the next GOTM rather than finish the last or simply because time runs out or my civ fanaticism for the month runs out before i finish. But its getting much better when i just connect my civ game to 1 game for 2 or 4 weeks. I imagine i might not be the only one who has this sort of issues with civ and plays GOTM for that reason primarily. As it is thus more than just a competition against other players, having a semi challenging difficulty level is something i like. Hence i suggested the handicap option because challenging can never be the same for different players.