Class F is pozzolanic only; Class C is pozzolanic and cementitious. Pozzolans require mixture with calcium hydroxide (present in portland cement) for hydration. Class C fly ash includes higher percentages of calcium oxide than Class F giving it cementitious properties independent of the portland cement. The difference is inconsequential since fly ash isn't used alone. Practically, the main difference is geography. In the U.S., Class C is more generally available in the western U.S., Class F the eastern U.S.
Generalizations are made about the effect of fly ash on concrete properties, but fly ash from different sources is highly variable in both classes. Its affect on various concrete properties is also highly variable depending on the specific fly ash used. Therefore, generalizations are helpful for determining what affects a particular fly ash might have, but the affect on concrete properties must be determined by trial batching.