Maybe here's a better, more intuitive explanation.
In one, we are looking for, # hands with at least two Aces / # hands with at least one Ace
and in two, we are looking for
# hands with at least two Aces (one of which is the Ace of Spades) / # hands with the Ace of Spades
In the second problem, the world in which you are operating is slightly different. There are fewer ways to have hands with 2 A's if one of them has to be the Spade A, and even though there are fewer hands with the Spade A, it doesn't cancel out perfectly.
It is very similar to that "two children one is a boy" vs "two children elder is a boy" question.