The two picks are dependent events. When you have to answer this kind of questions, you should ask yourself: "does knowledge about the first event influence what I expect from the second?"
If the answer is no, the two events are independent. Otherwise, they're dependent.
In this case, when you deal the first card, you have no information whatsoever. You expect each of the 52 cards to be dealt with the same probability 1/52.
Suppose you deal the ace of spades as the first card.
When you're about to deal the second card, does the knowledge about the first card dealt influence what you expect?
Yes! Even if you can't be sure about which card you will deal, you're SURE that you won't deal the ace of spade, because you didn't replace it.
So, the two events are dependent, because the first card dealt changed your expectations for the second card.
