[KyleGoetz]

I think 数える is limited to something like "1, 2, 3, 4, 5, 6, 7, 8...."
On the other hand 計算する can be used to count like that, but it additionally has the meaning of "calculate" as in "humm, 37 dollars per week times seven weeks, then income tax deduction of 20%....37*7*.8 is my post-tax income...and then divide that by (7*7*24) hours in that period of time...37*7*.8/(7*7*24) yields X dollars per hour."

計算する is all arithmetic and calculations. 数えるis more "to count."