The difference between the pair of inputs going into the s-box (consisting of two plaintexts xored with the first round key) is the same as the difference in the plaintexts alone. i.e. $(P_0 \oplus K_0) \oplus (P_1 \oplus K_0) = P_0 \oplus P_1$, because the round key $K_0$ cancels out. Hence, the input difference is independent of the value of the first round key. No matter what value you pick for $K_0$, the difference of the two inputs going directly into the s-box will be the same as the difference between the two plaintexts, with probability 1.
On the contrary, the output difference (difference between the outputs of the two s-boxes) is not independent of the round key. Or to be more precise, if you know $P_0$, $P_1$, and $K_0$ then the output difference just after the s-boxes -- $S(P_0 \oplus K_0) \oplus S(P_1 \oplus K_0)$ -- is fully determined and will vary depending on the precise value of $K_0$.
If, however, you assume that the pair of plaintexts is chosen uniformly at random from all pairs that satisfy some particular difference $\Delta P$, then the probability that the output difference will equal some particular value $\Delta C$ is independent of the precise value of $K_0$. This is because any given value for $K_0$ will simply exchange one pair of plaintexts that xor together to equal $\Delta P$ for another pair that xor together to equal $\Delta P$ (because of the equality in the first paragraph above). As such, the input pair going into the s-box after the roundkey is applied is still a uniformly random pair that xors together to equal $\Delta P$.
So under the assumption that the input difference is fixed but the input pair that xors together to equal that fixed difference is selected uniformly at random**, the probability that the output equals $\Delta C$ is purely a function of the (unkeyed) s-box, as summarized in the difference distribution table for the s-box. Note that this is different for key-dependent s-boxes.
** This assumption is equivalent to the assumption that the input pair is fixed but the roundkey is selected uniformly at random.