Score would seem to represent the ideal conditions that a designer has for their game. If everything is designed cohesively, a player with a higher score should be playing the game closer to it's pinnacle than one who isn't. This is VERY important in places where the player would be allowed to continue to play the game simply by inserting another quarter or continuing, because it usually severely separates these people from the folks who have 'mastered' the game.
That's pretty much it. All I care about is that the game has some way to acknowledge that the kid hogging the Neo Geo in front of me doesn't deserve to get on the scoreboard just because he insert a five into the token machine.
arcade games like the neo geo ones generally wiped your score to 0 on continue (and generally tell you how many times you died at the end), ones like the CAVE ones don't let you enter a high score if you used continues.