We study the asymptotic behavior of $\mathbb{P}(X-Y>u)$ as $u\to\infty$, where $X$ is subexponential and $X,Y$ are positive random variables that may be dependent. We give criteria under which the subtraction of $Y$ does not change the tail behavior of $X$. It is also studied under which conditions the comonotonic copula represents the worst-case scenario for the asymptotic behavior in the sense of minimizing the tail of $X-Y$ and an explicit construction of the worst-case copula is provided in the other cases.