I don't know of a common English word that includes both the idea of a teacher giving grades and a contest judge giving scores.
You used the word "judge" in your question, and in general someone who gives a score in a contest is called a "judge".
A person who keeps track of scores is called a "scorekeeper". But normally we don't use scorekeeper for someone who decides what your score should be, but just for the person who writes it down on a piece of paper or types it into the computer.
"Grader" is, I think, pretty rarely used. And in any case I've only heard it used for someone who grades school assignments, not for someone who assigns scores in a contest. Generally we just call the person who grades school assignments a "teacher". The title is understood to include the job of grading assignments.
There are words that you could use, like "assessor" or "evaluator". But the examples I can think of are obscure and have rather vague meanings. They could mean what you want, but they could also have other meanings depending on context.
If you need a word to include both ideas -- grading school papers and scoring contests --you could, of course, use an obscure word or make up a word. In either case, be sure to tell your readers what you are using it to mean.
Otherwise, you need to use a longer phrase, like "people who grade school assignments or judge contests".