Is there a word for a person who is very eager to earn points or scores? Not only in some academic field but everywhere, like sports or even if they join Stack Exchange, they start concentrating on getting reputation rather than on helping people.
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.
In cricket such a person is known as a statistician.
This is a common species, obsessed with the scores of everyone in every team, throughout history.
A true cricket statistician can recite the top ten scores by a number nine on a Saturday in Yorkshire since 1872. They are often found around the ground recording every ball's delivery and stroke. Then in a pub later discussing their favourite numbers.
And of course, they all worship at the altar of the great Bill Frindall. A legend amongst cricketers everywhere.