Difference between "an" and "one"

This is not English. It is mathematical logic. Your professor is trying to contrast the two statements:

(1) ∀ hour ∃ American who dies of melanoma.
(2) ∃ American who dies of melanoma ∀ hour.

As @Tim's comment suggests, the second statement means that the same American dies of melanoma every hour.

Neither of these statements is what either of the English sentences

(1) Almost every hour, an American dies of melanoma,
(2) One American dies of melanoma almost every hour,

really means in English. They both mean the same thing. If you want to state it mathematically, you need to say something like:

Consider the number of Americans n who die of melanoma between times t₁ and t₂, where the times are measured in hours. For large t₂ − t₁, n is with high probability slightly less than t₂ − t₁.