Ibn Ezra on Tehillim 147:4
סופר refers to the individual number, whereas מונה refers to the number in relation to the final count.
This would explain why it is called a מנין. We are counting each person in relation to the final total of 10. Less than that, doesn't mean anything, as we can't say Kaddish, etc. We need to reach the goal of 10.
ספירת העומר - We know that each day has its own counting. Apparently, it seems that each day has its own importance, and it seems to imply that each day when we count, we have a separate mitzvah. See this M.Y. question that I asked a while ago, regarding this concept.
Of curiosity of the use of סופר is Breishit 41:49 which uses this word in the context of Yosef counting how much grain he stored. The word makes sense, here, because he was trying to come up with an exact number. He didn't know what the total was beforehand. The verse says that "He gathered so much grain that he stopped counting because there was no number!" What does it mean to "not have a number?" See Siftei Chachamim #6 on this verse.