A self-avoiding, self-interacting polymer chain is studied both on a lattice and in the continuum using a Born-Green-Yvon integral equation approach. Equivalent theoretical approximations are made in both cases, allowing for an unambiguous comparison between the lattice and continuum models. The theory preserves the universal scaling behavior for polymer chain dimensions in the high-temperature limit and, with a lowering of temperature, predicts a universal collapse transition behavior for both lattice and continuum chains. Implications for the modeling of polymer solutions are discussed.