The Rank you get primarily impacts the GMP (Gross Military Product) you receive. I created a small table that shows the different ranks, their scores and their bonus.
Rank Score Bonus
S 130,000 20%
A 100,000 18%
B 60,000 20%
C 30,000 30%
D 10,000 45%
E <10,000 100%
This means that if you have a score of 130 000 you'd get a GMP of 26000 (130000*0.2) while having a score o 10000 would net you 4 500 (10000*0.45) and 9999 translate to 9999 (9999*1). This boosts you if you are performing bad yet places quite a limitation on the GMP you receive.
Of course there also is the fact that there are Achievements for acquiring S-Rank, but those don't really have much impact on the game.