A game is a game. It's software entertainment that involves immersion into a world that offers more interactivity than other forms of entertainment, but lacks commitment or serious tone to how the game is played. People who play games seriously or declare themselves as pro aren't really fooling anybody, nobody worthwhile at least.
A non-game is something that simulates real life in a certain aspect, but simplifies or cheapens the experience into a digestible and low-cost entertainment medium, and since video games in certain cases require the amount of interactivity simply living your life, the people who create these simulations use video games because the gap left is shortened.
Games like the Sims, Nintendogs, Flight Simulator, and other games of a similar genre are not in fact games since the gameplay consists of, more or less, what you'd end up doing if you were to actually try the simulation in real life. Regardless of how real some simulations attempt to be, we still can't use them as an applicable form of training.
Having hours of Flight Simulator X isn't going to get you on the fast track to touring the sky anytime soon. Therefor we still classify them as games, just as non-games because it's simulating something we can do without video games.
Brain-Age is an example of a game comprised of simple thinking tasks we may or may not actually preform in real life, but the game does a great job of simulating it.
A video game is like a movie of drama, action, romance, comedy, etc. A non-game is comparable to a documentary, lecture or something similar.
Regardless, that's the distinction between games and non-games, at least the way I see it.