In a recent paper, Joe and Zhu [2005, Generalized Poisson distribution: the property of mixture of Poisson and comparison with negative binomial distribution. Biometrical Journal, 47 (2), 219–229.] compared the negative binomial and the generalized Poisson distributions. We aim at extending this comparison by including more distributions and more aspects, like tailness and kurtosis. We also aim at describing the related problem, whether from real data one may distinguish between the candidate models. To do this, we consider several aspects of three of the most commonly used mixed Poisson distributions, namely the negative binomial, the Poisson inverse Gaussian and the generalized Poisson distributions. The results show that for small mean and overdispersion, all the models are quite the same, whereas for larger means the generalized Poisson and the Poisson inverse Gaussian distributions have larger tails than the negative binomial, and the differences are much larger. In practice, it is not easy to discriminate between them for small counts and small overdispersion, but for large overdispersion discrimination is relatively easy. Applications to real data are provided to illustrate the ideas.