I understand that at static equilibrium, the electric field inside a conductor is zero, so the potential difference is zero between any two points within the conductor. However, the surface of a charged solid spherical conductor has a net charge, and thus a net electric field, unlike the inside of the sphere. So why is the electric potential within the conductor and the surface of the conductor the same?
The field is actually discontinuous at the surface: the discontinuity in the field is proportional to the surface charge density.
The statement "within the conductor and the surface" is to be understood as meaning within the conductor and a point arbitrary close to the surface but inside this surface. The situation you describe is an idealization as, in real conductors, the charge is concentrated in a small boundary around the surface; the thickness of this boundary depends inversely on the conductivity of the material, and goes to zero in the ideal case of a perfect conductor with conductivity $\sigma\to\infty$.
The fact that there is an electric field outside the conductor only means that the potential will change more the further away you go from the surface. The field can be thought of as a derivative (or more precisely, a gradient) of the potential with respect to distance. The potential will change starting from the potential of the conductor at the surface, and then grow or decrease as you get further away from the conductor.