We present a detailed mass reconstruction and a novel study on the substructure properties in the core of the Cluster Lensing And Supernova survey with Hubble (CLASH) and Frontier Fields galaxy cluster MACS J0416.1-2403. We show and employ our extensive spectroscopic data set taken with the VIsible Multi-Object Spectrograph instrument as part of our CLASH-VLT program, to confirm spectroscopically 10 strong lensing systems and to select a sample of 175 plausible cluster members to a limiting stellar mass of log (M ∗/M ) ≃ 8.6. We reproduce the measured positions of a set of 30 multiple images with a remarkable median offset of only 0.″3 by means of a comprehensive strong lensing model comprised of two cluster dark-matter halos, represented by cored elliptical pseudo-isothermal mass distributions, and the cluster member components, parameterized with dual pseudo-isothermal total mass profiles. The latter have total mass-to-light ratios increasing with the galaxy HST/WFC3 near-IR (F160W) luminosities. The measurement of the total enclosed mass within the Einstein radius is accurate to 5%, including the systematic uncertainties estimated from six distinct mass models. We emphasize that the use of multiple-image systems with spectroscopic redshifts and knowledge of cluster membership based on extensive spectroscopic information is key to constructing robust high-resolution mass maps. We also produce magnification maps over the central area that is covered with HST observations. We investigate the galaxy contribution, both in terms of total and stellar mass, to the total mass budget of the cluster. When compared with the outcomes of cosmological N-body simulations, our results point to a lack of massive subhalos in the inner regions of simulated clusters with total masses similar to that of MACS J0416.1-2403. Our findings of the location and shape of the cluster dark-matter halo density profiles and on the cluster substructures provide intriguing tests of the assumed collisionless, cold nature of dark matter and of the role played by baryons in the process of structure formation.