Not a mathematician, but AIUI, the average value pertains to some real, finite set of cases, whereas the expected value pertains to the abstract set of cases.
As in, you roll 2d6 ten times, and get 6, 9, 2, 4, 10, 7, 2, 12, 7 and 9. The average here is 6.8, when the expected value is 7. If you roll perfectly fair dice infinitely many times, it will converge to 7.
