What I DON'T like about scoring systems...
They are potentially pointless (pun intended).
It can be distracting, and take you out of the story, and remind you you're playing a game.
What I DO like about scoring systems...
Some games have different endings or situations that happen, depending on your score. In those cases, the system isn't so pointless.
The old Sierra Camelot game was like that, I believe. The score gave you a heads-up as far as whether you would get the "good" ending.
Scoring system also alerts you to any puzzles you missed, assuming it's possible to do that and still progress to the end.
It can also alert the gamer as to how far along they are, but I rather the story do that.
Personally, unless I'm playing Atari where there is no end, and score is ALL there is, I don't care about score. And I would imagine a lot of gamers are like that.
I would use scoring if there were multiple ways to solve a puzzle, and it would be a way to "reward" and persuade gamers to figure out the "best" way to do it,
knowing they would get more points.
Otherwise I don't see a point to having scoring. They either solve it or don't. And the true reward for solving it is typically they get to progress through the story.