The atomic and electronic structure of the lattice-mismatched h-BN/Pt͑111͒ and h-BN/Rh͑111͒ interfaces formed by pyrolitic reactions with vaporized borazine has been studied by low-energy electron diffraction, scanning tunneling microscopy, x-ray-absorption spectroscopy, and core-level and valence-band photoemission. It has been found that on Pt͑111͒, h-BN forms a nearly flat monolayer, insignificantly corrugated across the supercell. On Rh͑111͒, h-BN grows in form of a nanomesh, as originally observed by Corso et al. ͓Science 303, 217 ͑2004͔͒. The structural difference between the h-BN/Pt͑111͒ and h-BN/Rh͑111͒ interfaces is associated with the strength of chemical interaction between h-BN and the substrate surface. A stronger orbital hybridization on Rh͑111͒ results in a stronger attraction of the monolayer to the metal surface at favorable adsorption sites resulting in a highly corrugated structure ͑nanomesh͒. It has been shown that the electronic structure of the outer ͑elevated͒ and inner ͑attracted to the surface͒ nanomesh sites is very different as a result of different chemical bonding to the substrate ͑weak and strong, respectively͒.