To address the OP, RPGs are different because they are much more RP than G. When I play D&D, I am pretending to be a wizard or fighter or whatever. This is not much different from my daughter pretending to be a kangaroo who runs a restaurant, and not at all like playing baseball or monopoly or even Descent. The "game" part is a framework for playing pretend, so several of us can do together.
In a regular game, each player or team is trying to win, and there are rules so we know who wins. In that case, you wouldn't want to change rules on the fly (exception based on how informal the game is). In RPGs, the players aren't trying to beat each other, or even trying as a team to beat the DM, they're just all there having fun pretending to be knights or wizards or superheroes. The rules aren't there to make it a fair contest between the DM and players, they are there so we can all be on the same page about what we are pretending. The "game" is designed for the DM to make judgment calls over when to rigidly apply the rules and when to just decide based on coolness or verisimilitude or not wanting to kill someone with a freak series of rolls. At heart, its a role-playing activity with some rules, not a game with large number options.