Correlations - Get values in the way we want
Several kernel functions can serve as similarity functions (=scores). See a list, for example, here. You can try several of them and see which suits you the best.

You need something that drops fast at low distances. You can try
$$ score = 1/(1+distance)^2$$
and adjust coefficient in front of distance so that the score fits between 0 and 1
  

About your picture: what are axis labels? and what are x-ticks?