The answer is $\mathbf{(B)}$. $x^{2} - y^{2}$ can be factored as $(x + y)(x - y)$. It is not possible for $x + y$ and $x - y$ to both be multiples of $4$, because otherwise $(x + y) + (x - y) = 2x$ would also be, but this is twice an odd number. We have that $\max\{x + y, x - y\} \leq 2011 + 2011 < 2^{12}$, therefore the maximum power of $2$ obtainable in the factorization of $x^{2} - y^{2}$ is $2^{12}$; this is achieved for example by setting $x + y = 2^{11} = 2048$ (the next power is too large) and $x - y = 2$, that is $x = 1025$ and $y = 1023$.