Is there a word for person who is obsessed with scores

Solution 1:

The most accurate term is point-whore. This is a term common to tabletop gaming and computer gaming. This type of person only plays in order to get the most points or highest score in the game. Often this type of player even sabotages the game in order to end up with more points than the others.

Solution 2:

Pathologically competitive would be my description of such a person, but that is two words and you asked for "a word."

As far as "not being competitive while wants to have more points" -- more points than whom? If he is not competing, the points wouldn't mean anything. It's obviously a competition in his mind.