In this paper, we derive two simple and asymptotically exact approximations for the function defined as $\Delta{\cal Q}_{m}(a,\, b)\buildrel{\Delta}\over{=}Q_{m}(a,\break \, b)-Q_{m}(b,\,a)$ . The generalized Marcum $Q$-function $Q_{m} (a,\, b)$ appears in many scenarios in communications in this particular form and is referred to as the symmetric difference of generalized Marcum $Q$-functions or the difference of generalized Marcum $Q$-functions with reversed arguments. We show that the symmetric difference of Marcum $Q$-functions can be expressed in terms of a single Gaussian $Q$-function for large and even moderate values of the arguments $a$ and $b$. A second approximation for $\Delta{\cal Q}_{m} (a,\, b)$ is also given in terms of the exponential function. We illustrate the applicability of these new approximations in different scenarios: 1) statistical characterization of Hoyt fading; 2) performance analysis of communication systems; 3) level crossing statistics of a sampled Rayleigh envelope; and 4) asymptotic approximation of the Rice $I_{e}$-function.