We have performed a systematic semi-empirical and ab initio van der Waals study to investigate the bonding mechanism of benzene (C(6)H(6)), triazine (C(3)N(3)H(3)) and borazine (B(3)N(3)H(6)) adsorbed on graphene and a single boron nitride (BN) sheet. The two semi-empirical approaches used to include the van der Waals (vdW) interactions in our density functional theory (DFT) calculations suggest that the strength of the molecule-surface interaction corresponds to a strong physisorption with no net charge transfer between the molecules and the corresponding substrates. This observation is strengthened by the use of first-principles non-local correlation vdW-DF functionals which provide a sound physical basis to include vdW interactions in DFT calculations. In particular we have employed two flavors of vdW-DF functionals which enabled us to determine the role of the non-local correlation effects in the molecule-surface bonding mechanism which cannot be assessed by using only semi-empirical vdW methods. Our study also reveals that the strength of the molecule-surface interaction can be influenced by the electronegativity of the B, C and N atoms.