We experimentally investigate the modifications of wall frictional drag and mean velocity profile by the viscoelasticity of fluid in high Reynolds-number Taylor- Couette turbulence both for counter-rotation and co-rotation of the inner and outer cylinders of a very large facility. As working fluids, water and a dilute surfactant solution are examined. We conduct systematic measurements of the torque on the inner cylinder with varying the angular velocity ratio of the cylinders and the inner Reynolds number. Torque measurement shows that remarkable drag enhancement occurs due to the viscoelasticity of the solution in a relatively lower Reynolds number regime, especially for the co-rotation and the pure inner cylinder rotation, as reported by the previous direct numerical simulations for the pure inner cylinder rotation. On the other hand, drag reduction is observed at a sufficiently high Reynolds number for the counter-rotation. By defining a new Reynolds number based on the bulk shear rate, these results imply that we may understand the drag modification with respect to the time-scale of the viscoelasticity both for co- and counter-rotation in a unified manner Concerning the modification of the mean azimuthal velocity profile, a remarkable quasi-solid-body rotational flow is sustained in a bulk region for the drag enhancement cases whereas, for the drag reduction cases, the mean flow structure seems to be modified only in the vicinity of the wall. The rotational angular velocity of the quasi-solid-body rotational flow shows a high correlation with the drag reduction ratio.