It is well known that superconductivity in thin films is generally suppressed with decreasing thickness. This suppression is normally governed by either disorder-induced localization of Cooper pairs, weakening of Coulomb screening, or generation and unbinding of vortex-antivortex pairs as described by the Berezinskii-Kosterlitz-Thouless (BKT) theory. Defying general expectations, few-layer NbSe2 - an archetypal example of ultrathin superconductors - has been found to remain superconducting down to monolayer thickness. Here we report measurements of both the superconducting energy gap and critical temperature in high-quality monocrystals of few-layer NbSe2, using planar-junction tunneling spectroscopy and lateral transport. We observe a fully developed gap that rapidly reduces for devices with the number of layers N < 5, as does their ctitical temperature. We show that the observed reduction cannot be explained by disorder, and the BKT mechanism is also excluded by measuring its transition temperature that for all N remains very close to Tc. We attribute the observed behavior to changes in the electronic band structure predicted for mono- and bi- layer NbSe2 combined with inevitable suppression of the Cooper pair density at the superconductor-vacuum interface. Our experimental results for N > 2 are in good agreement with the dependences of the gap and Tc expected in the latter case while the effect of band-structure reconstruction is evidenced by a stronger suppression of the gap and the disappearance of its anisotropy for N = 2. The spatial scale involved in the surface suppression of the density of states is only a few angstroms but cannot be ignored for atomically thin superconductors.