Experimental and theoretical investigations of the K-shell X-ray absorption near edge structure (XANES) in hexagonal and cubic boron nitride crystals were carried out. A central role is revealed of: (1) the atomic B and N 1s^-1 2p resonances in the formation of the main resonance features, (2) the splitting of these atomic excitations by an anisotropic surroundings potential into single π and double degenerated a components in hexagonal relative to cubic BN crystal in which they remain triple degenerated, and (3) the localization region size determining the essential difference between the B and N K-shell excitations. The quasi-atomic approach to inner-shell photoprocesses is applied to compute the XANES. The applicability of molecular models for simulation of the core excitations in the crystals is discussed. It is shown that the B K-excitations are essentially localized within the nearest neighbors whereas the N K ones are more delocalized. The experimental measurements with high energy resolution evidence that the full width at half maximum for the low-lying excitation in h-BN is ≈ 0.37 eV and in its vicinity a near-π-resonance structure containing a shoulder at 192.3 eV and a well-resolved narrow peak at 193.1 eV appears. Its origin is rationalized taking into account the local atomic rearrangement allowing the out-of-plane displacements of excited-core B atoms from their regular positions (polaron mechanism).