The object of experimental study is a fluid flow generated by differential rotation of a free light spherical body in a rotating cylindrical cavity. The body stays near the axis under the action of centrifugal force. The body rotation is generated by a force field oscillating in the cavity reference system (vibrational hydrodynamic top). It was found that the Taylor-Proudman column that forms undergoes instability, which manifests itself in the formation of a two-dimensional azimuthal wave at the column boundary, in a Stewartson layer. The experimental results are summarized on a plane of dimensionless parameters, i.e., the dimensionless velocity of the cavity rotation and Rossby number. The bounds of the Stewartson layer stability were found and the supercritical structures and transition sequences were studied. Systematic research into that problem in its classical formulation--when a sphere is fixed on the axis and its differential rotation is imposed--was done for comparison. It was demonstrated that in conditions of vibratory differential rotation of a free sphere the stability threshold of the Stewartson layer was reduced by more than one order of magnitude, in comparison with the classical case. A qualitative change was also found in the wave phase velocity which for a free sphere exceeds the lagging differential rotation velocity of the body. It was concluded that the uncovered specifics are related to the difference in the mechanism of the Taylor-Proudman column formation and of the flow generation in it. For a vibrational hydrodynamic top, streams in the column will not be defined by Ekman pumping but by steady streaming, which is also responsible for the free-sphere differential rotation.