The adsorption of H2O, NH3 and HCOOH as polar probe molecules and C6H6 and CH4 as non-polar ones on a series of zig-zag (n,0) single-walled boron nitride nanotubes (BNNTs) and on a boron-nitride mono-layer (BNML) has been studied by means of B3LYP-D* periodic calculations. Computed electrostatic potential maps for the pristine BN nanomaterials indicate that the smaller the radius, the larger the polar character. Polar molecules are found to be strongly chemisorbed on small radius BNNTs by means of dative interactions between electron donor atoms of the molecules and B atoms of the BNNTs, H-bonding, as well as dispersive forces. Remarkably, for HCOOH interacting with the (4,0) BNNT, this dative interaction is accompanied by a proton transfer to the nanotube. The corresponding computed adsorption energies decrease sharply with increasing tube radius, gradually approaching the values for physisorption on the BNML. Adsorption of non-polar molecules, mainly dictated by π-stacking (C6H6) and CH-π (CH4) dispersion interactions, is found to be energetically more favorable when physisorbed on large radius BNNTs, the most stable adducts being formed on the BNML.