This randomized phase III trial studies pemetrexed disodium to see how well it works compared with erlotinib hydrochloride as second-line therapy in treating patients with non-small cell lung cancer that has spread to other places in the body. Pemetrexed disodium and erlotinib hydrochloride may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. It is not yet known whether pemetrexed disodium is more effective than erlotinib hydrochloride in treating advanced non-small cell lung cancer.