$\because |x-y|=y-x$，

$\therefore y \gt x$，

$\because |x|=3$，$|y|=4$，

$\therefore x=\pm 3$，$y=4$，

$\therefore \left(x+y\right)^{3}=\left(3+4\right)^{3}=343$；

或$\left(x+y\right)^{3}=\left(-3+4\right)^{3}=1$.

$\therefore \left(x+y\right)^{3}=343$或$1$.

$\because |x-y|=y-x$，

$\therefore y \gt x$，

$\because |x|=3$，$|y|=4$，

$\therefore x=\pm 3$，$y=4$，

$\therefore \left(x+y\right)^{3}=\left(3+4\right)^{3}=343$；

或$\left(x+y\right)^{3}=\left(-3+4\right)^{3}=1$.

$\therefore \left(x+y\right)^{3}=343$或$1$.