We analyze an extended redshift sample of Abell/ACO clusters and compare the results with those coming from numerical simulations of the cluster distribution, based on the truncated Zel'dovich approximation (TZA), for a list of eleven dark matter (DM) models. For each model we run several realizations, so that we generate a set of 48 independent mock Abell/ACO cluster samples per model, on which we estimate cosmic variance effects. Other than the standard CDM model, we consider (a) Ω0 = 1 CDM models based on lowering the Hubble parameter and/or on tilting the primordial spectrum; (b) Ω0 = 1 Cold + Hot DM models with 0.1 ≤ Ων ≤0.5; (c) low-density flat ΛCDM models with 0.3 ≤ Ω0 ≤0.5. We compare real and simulated cluster distributions by analysing correlation statistics, the probability density function, and supercluster properties from percolation analysis. We introduce a generalized definition of the spectrum shape parameter Γ in terms of σ25/ σ8, where σ ris the rms fluctuation amplitude within a sphere of radius r. As a general result, we find that the distribution of galaxy clusters provides a constraint only on the shape of the power spectrum, but not on its amplitude: a shape parameter 0.18 ≲ Γ ≲ 0.25 and an effective spectral index at the 20 h-1 Mpc scale -1.1 ≲ neff ≲ -0.9 are required by the Abell/ACO data. In order to obtain complementary constraints on the spectrum amplitude, we consider the cluster abundance as estimated using the Press-Schechter approach, whose reliability is explicitly tested against N-body simulations. By combining results from the analysis of the distribution and the abundance of clusters we conclude that, of the cosmological models considered here, the only viable models are either Cold + Hot DM ones with 0.2 ≲ Ων ≲ 0.3, better if shared between two massive ν species, and ΛCDM ones with 0.3 ≲ Ω0≲0.5.