We present an extended version of the 2-phase gas-grain code NAUTILUS to the 3-phase modelling of gas and grain chemistry of cold cores. In this model, both the mantle and the surface are considered as chemically active. We also take into account the competition among reaction, diffusion and evaporation. The model predictions are confronted to ice observations in the envelope of low-mass and massive young stellar objects as well as toward background stars. Modelled gas-phase abundances are compared to species observed toward TMC-1 (CP) and L134N dark clouds. We find that our model successfully reproduces the observed ice species. It is found that the reaction-diffusion competition strongly enhances reactions with barriers and more specifically reactions with H2, which is abundant on grains. This finding highlights the importance to have a good approach to determine the abundance of H2 on grains. Consequently, it is found that the major N-bearing species on grains go from NH3 to N2 and HCN when the reaction-diffusion competition is accounted. In the gas-phase and before few 10^5 yrs, we find that the 3-phase model does not have a strong impact on the observed species compared to the 2-phase model. After this time, the computed abundances dramatically decrease due to the strong accretion on dust, which is not counterbalanced by the desorption less efficient than in the 2-phase model. This strongly constrains the chemical-age of cold cores to be of the order of few 10^5 yrs.