There's a few leaderboards for Fallout Shelter - one of them is for "score." How is that score calculated?
Your "Score" in the leaderboards is determined by the Achievements. Since each achievement has a certain amount of "Experience"/"Points" (Android/iOS respectively), depending on which achievement(s) you have, determines your overall "Score" for the game.
For example, a person with 2 achievements worth 100 points each (200 total), would have a better "Score" than a person with 3 achievements worth only 50 each (150 total).