We have studied the nucleation and growth of cubic boron nitride (cBN) films deposited on silicon and diamond-coated silicon substrates using fluorine-assisted chemical vapor deposition (CVD). These comparative studies substantiate that the incubation amorphous/turbostratic BN layers, essential for the cBN nucleation on silicon, are not vital precursors for cBN nucleation on diamond, and they are inherently eliminated. At vastly reduced critical bias voltage, down to -10 V, cBN growth is still maintained on diamond surfaces, and cBN and underlying diamond crystallites exhibit an epitaxial relationship. However, the epitaxial growth is associated with stress in the cBN-diamond interfacial region. In addition, some twinning of crystallites and small-angle grain boundaries are observed between the cBN and diamond crystallites because of the slight lattice mismatch of 1.36%. The small-angle grain boundaries could be eliminated by imposing a little higher bias voltage during the initial growth stage. The heteroepitaxial growth of cBN films on different substrate materials are discussed in the view of lattice matching, surface-energy compatibility, and stability of the substrate against ion irradiation.