The easy, hard and split core configurations of the <111> screw dislocation and the energy pathways between them are studied in body-centered cubic (bcc) Fe and W using different density functional theory (DFT) approaches. All approaches indicate that in Fe, the hard core has a low relative energy, close to or even below that of the saddle configuration for a straight path between two easy cores. This surprising result is not a direct consequence of magnetism in bcc Fe. Moreover, the path followed by the dislocation core in the (111) plane between easy cores, identified here using two different methods to locate the dislocation position, is almost straight, while the energy landscape between the hard core position and the saddle configuration for a straight path is found to be very flat. These results in Fe are in contrast with predictions from empirical potentials as well as DFT calculations in W, where the hard core has an energy about twice that of the maximum energy along the Peierls barrier, and where the dislocation trajectory between easy cores is curved. Also, the split core configuration is found to be unstable in DFT and of high energy in both Fe and W, in contrast with predictions from most empirical potentials.