Why should keys be hidden from an eavesdropper for a pseudorandom function? What advantage would knowing the key used for a given pseudorandom function give over having access to an oracle for that function $F_k(\cdot)$ and the actual key $k$ which will be used?

Thanks very much in advance
With $k$ the adversary can compute $F_k(x)$ for some $x$ using the evaluation algorithm, and compare it to the answer of its oracle on $x$.