A notable characteristic of PbTaSe$_2$, a prototypical noncentrosymmetric (NCS) superconductor, is that its superconductivity can be modulated through a structural transition under hydrostatic pressure [Phys. Rev. B 95, 224508 (2017)]. Here we report on simultaneous pressure-sensitive point-contact Andreev reflection (PCAR) spectroscopy and bulk resistance measurements on PbTaSe$_2$, to elucidate the nature of the surface and bulk superconductivity and their evolution with hydrostatic pressure. It is found that in high pressure region the superconducting gap opening temperature $T_c^A$ is significantly lower that the bulk resistive transition temperature $T_c^R$, revealing a clear experimental signature of surface-bulk separation associated with enhanced antisymmetric spin-orbit coupling (ASOC). The PCAR spectra, reflecting the superconducting surface state, are analyzed with the Blonder-Tinkham-Klapwijk theory, yielding an isotropic $s$-wave full BCS-gap in the strong coupling regime. Analysis based on a modified McMillan formula indicates a sizable coupling strength contributed from ASOC for the superconducting surface state. These results suggest the coexistence of full gap $s$-wave superconductivity and topological surface states in PbTaSe$_2$, indicating that this NSC with significantly enhanced ASOC may offer a solid platform to investigate the topological aspect in the superconducting condensate.
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