Skew elliptical copula models are attractive for modelling economic and financial data because they allow for asymmetric and tail dependence. We consider two flexible classes of skew-elliptical copulas which have semi-closed-form copula densities and can be readily applied in high-dimensional applications. The first class is the skewed generalized hyperbolic (GH) copula implied by a multivariate normal location-scale mixture with generalized inverse Gaussian (GIG) mixing distribution, and the second class is the modulated skew-elliptical copula implied by a multivariate modulated skew-elliptical distribution. Both classes of models are characterized by a skewness vector, a generalized correlation matrix, and certain tail thickness parameters. However, we show that these two class of skew-elliptical copulas have distinct properties in terms of pairwise tail dependence properties and rank dependence measures. For the pairwise tail dependence, the second class of copulas yield nontrivial asymptotic dependence in the extreme values taken in all four quadrants, while the first class of copulas imply asymptotic tail independence in almost all the cases. For rank dependence, we derive the explicit formula for the Kendall’s tau and Spearman’s rho coefficients and demonstrate that these rank correlation coefficients of the two models deviate from the symmetric elliptical case in very different manners.