How is a roleplaying game not a board game?
To give a short, simple attempt at answer to kick things off: I would say that a roleplaying game must be partly comprised of hidden information that has been invented by one or more of the players.  

That's the key difference I can see between The Arkham Horror/Mansions of Madness (boardgames) and The Call of Cthulhu (roleplaying game), where both are moderated by a Keeper.  In both setups the Keeper controls the opposition to the players, but in only one is (s)he asked to invent details, descriptions or outcomes, as opposed to just managing them.

In a boardgame, it would be seriously frowned upon for any player (even in a moderatorial role) to come up with new challenges or house rules for the others that are not in the script.  In a roleplaying game, that's practically compulsory.