We study the linear and nonlinear structure formation in the dilaton and symmetron models of modified gravity using a generic parameterisation which describes a large class of scenarios using only a few parameters, such as the coupling between the scalar field and the matter, and the range of the scalar force on very large scales. For this we have modified the N-body simulation code ECOSMOG, which is a variant of RAMSES working in modified gravity scenarios, to perform a set of 110 simulations for different models and parameter values, including the default ΛCDM. These simulations enable us to explore a large portion of the parameter space. We have studied the effects of modified gravity on the matter power spectrum and mass function, and found a rich and interesting phenomenology where the difference with the ΛCDM template cannot be reproduced by a linear analysis even on scales as large as k ~ 0.05 h/Mpc-1. Our results show the full effect of screening on nonlinear structure formation and the associated deviation from ΛCDM. We also investigate how differences in the force mediated by the scalar field in modified gravity models lead to qualitatively different features for the nonlinear power spectrum and the halo mass function, and how varying the individual model parameters changes these observables. The differences are particularly large in the nonlinear power spectra whose shapes for ƒ(R), dilaton and symmetron models vary greatly, and where the characteristic bump around 1 h/Mpc-1 of ƒ(R) models is preserved for symmetrons, whereas an increase on much smaller scales is particular to symmetrons. No bump is present for dilatons where a flattening of the power spectrum takes place on small scales. These deviations from ΛCDM and the differences between modified gravity models, such as dilatons and symmetrons, could be tested with future surveys.