The best example I can think of is Civilization II. (I've bought copies of Civ: CTP and Civ 3, but never got round to playing them - Civ II is the one I've played the most.) When I was living with William, I noticed that we had very different styles of gameplay. My approach was to be the best leader I could be, so I was looking out for the best interests of my people. William's goal was to get the highest score he could at the end of the game (I'm sure he'll jump in here if I'm mis-representing him!). So, for instance, when it got up to the endgame, one thing you can do is sell off all your buildings. That gives you extra gold, which translates to extra points, but it does mean that your citizens will start dying on the streets shortly afterwards. If the game has finished by then, then it wont affect your score, so you're quids in. However, I refused to use that tactic on principle - I wouldn't sacrifice the kingdom's long-term future for my own short-term benefit. And yes, I'm aware it's not real, but I figure that you have to suspend your disbelief a bit, otherwise there's no point playing. Mind you, before you start thinking I should go into politics, I've tended to play as monarch rather than president, since that way I can make all the decisions unilaterally, without having to worry about the hoi polloi out-voting me :)
There's an old TNG episode, where Worf says (in reference to sport):
"If winning is not important, why keep score?"
In the case of Civ, the answer is that I don't. I often keep playing past the "final turn" - you can keep going indefinitely, but your score is frozen. In other computer games, I'm not actually keeping score myself - the computer does it for me, and I don't have any control over that. I suppose you have to keep score with Magic, since life points are an intrinsic part of the game, but I can see the appeal of games like Munchkin, where you never actually get knocked out completely.
I also quite like the idea of people getting handicaps, to level the playing field a bit. E.g. when I used to play chess, I would sometimes start out without any rooks. That gave my opponent a better chance, which was good for zir, and it forced me to improve my knight/bishop strategy, which was good for me.