The “full”-Stokes ice flow model is now commonly used as the metric of “truth” for verifying lower-order glacier and ice sheet models. Mountain glaciers, generally located in remote and difficult to access regions, are often hard to simulate due to a lack of necessary model input data, most specifically accurate information on glacier geometry. For this reason, it is often convenient to measure glacier geometry only along a central flowline and to model evolution of those glaciers using a 2D flowline model with parameterizations for capturing across-flow geometric effects. To date, the accuracy and applicability of such flowline models has still not been fully investigated. Here, we systematically study the applicability of a 2D, first-order Stokes approximation flowline model, modified by geometric shape factors, for the simulation of land-terminating glaciers by comparing it with a 3D, “full”-Stokes ice-flow model. We explore the sensitivities of the flowline and Stokes models to ice geometry, temperature, and forward model integration time using steady-state and transient, thermomechanically uncoupled and coupled numerical experiments. Our findings show that the 2D, first-order flowline model may produce inaccurate results for (1) steep glaciers with complex basal topography, (2) polythermal glaciers that contain temperate basal ice and experience basal sliding, and (3) coupled thermomechanical glacier evolution over long time periods (~103 years). We conclude that the 2D first-order flowline model should be applied and interpreted with caution when modeling glacier changes under a warming climate or over long periods of time.