We study the morphological content of a large sample of high-redshift clusters to determine its dependence on cluster mass and redshift. Quantitative morphologies are based on PSF-convolved, 2D bulge+disk decompositions of cluster and field galaxies on deep Very Large Telescope FORS2 images of eighteen, optically-selected galaxy clusters at 0.45 < z < 0.80 observed as part of the ESO Distant Cluster Survey ("EDisCS"). Morphological content is characterized by the early-type galaxy fraction f(et), and early-type galaxies are objectively selected based on their bulge fraction and image smoothness. This quantitative selection is equivalent to selecting galaxies visually classified as E or S0. Changes in early-type fractions as a function of cluster velocity dispersion, redshift and star-formation activity are studied. A set of 158 clusters extracted from the Sloan Digital Sky Survey is analyzed exactly as the distant EDisCS sample to provide a robust local comparison. We also compare our results to a set of clusters from the Millennium Simulation. Our main results are: (1) the early-type fractions of the SDSS and EDisCS clusters exhibit no clear trend as a function of cluster velocity dispersion. (2) Mid-z EDisCS clusters around sigma = 500 km s(-1) have f(et) similar or equal to 0.5 whereas high-z EDisCS clusters have f(et) similar or equal to 0.4. This represents a similar to 25% increase over a time interval of 2 Gyr. (3) There is a marked difference in the morphological content of EDisCS and SDSS clusters. None of the EDisCS clusters have early-type galaxy fractions greater than 0.6 whereas half of the SDSS clusters lie above this value. This difference is seen in clusters of all velocity dispersions. (4) There is a strong and clear correlation between morphology and star formation activity in SDSS and EDisCS clusters in the sense that decreasing fractions of [OII] emitters are tracked by increasing early-type fractions. This correlation holds independent of cluster velocity dispersion and redshift even though the fraction of [OII] emitters decreases from z similar to 0.8 to z similar to 0.06 in all environments. Our results pose an interesting challenge to structural transformation and star formation quenching processes that strongly depend on the global cluster environment (e. g., a dense ICM) and suggest that cluster membership may be of lesser importance than other variables in determining galaxy properties.