Consider the following example. $W, X, Y, Z$ all have standard Cauchy distribution, but $X$ and $Y$ are independent while $Z = -W$. Then $X+Y$ also has a Cauchy distribution so its expected value does not exist, while $W+Z=0$ identically.

Consider the following example. $W, X, Y, Z$ all have standard Cauchy distribution, but $X$ and $Y$ are independent while $Z = -W$. Then $X+Y$ also has a Cauchy distribution so its expected value does not exist, while $W+Z=0$ identically.

EDIT: With the additional assumption that $E[X+Y]$ exists, I believe you'll find that $$ E[X+Y] = \int_{-\infty}^\infty (F_{-X}(t) - F_{Y}(t))\ dt $$ (where $F_{-X}$ and $F_Y$ are the cumulative distribution functions of $-X$ and $Y$), so that this is the same as $E[W+Z]$.