Optical surface modes are specific states of electromagnetic waves localized at the interface separating two dissimilar media where the wave vector becomes complex causing the wave to exponentially decay away from the surface. These general conditions permit surface modes to form in a wide range of systems including layered optical media, optical waveguides, metallic thin films, carbon nano-tubes, and photonic crystals. Equally remarkable are the effects based on surface modes, such as extraordinary optical transmission through subwavelength apertures and beaming of light. In this paper, we analyse the surface modes, also known as Tamm states for electronic systems, along two surface orientations of a semi-infinite binary photonic crystal formed by a square lattice of high dielectric rods in vacuum. We reveal the conditions required to form localised surface modes in this system without perturbation of the surface layer, such as a reduction in the surface rod radius or refractive index. In this way, we demonstrate the existence of intrinsic surface modes at a photonic crystal surface. In addition to the study of linear surface states, we introduce a third-order optical nonlinearity to the surface layer and analyse the properties of the nonlinear surface Tamm states. We investigate the energy threshold, dispersion, and modal symmetries of the surface states, and illustrate their nonlinearity-induced tunability.