Competing and Complementary risk (CCR) problems are often modelled using a class of distributions of the maximum, or minimum, of a random number of i.i.d. random variables; we call this class the CCR class of distributions. While the CCR distributions generally do not have an easy-to-calculate density or probability mass function, two special cases, namely the Poisson-exponential and the exponential geometric distributions, can easily be calculated. Hence, it is of interest to approximate CCR distributions with these simpler distributions. In this paper, we develop Stein's method for the CCR class of distributions to provide a general comparison approach to bound the distance between two CCR distributions and contrast this approach to bounds obtained using a Lindeberg argument. We detail the comparison for Poisson-exponential and exponential-geometric distributions.