The difference between the two processes which result in viscosity is down to how molecules interact with one another.
In a gas the interactions between molecules, which are relatively far apart, are primarily via collisions and the "nearest neighbours$ change all the time.
In a liquid the the interactions between the molecules, with a spacing roughly the size of a molecule, are via bonds between the molecules and although nearest neighbours do change that happens relatively slowly as compared to molecules in a liquid.
The different mechanisms are described in the Physical Causes section of the Wikipedia article Viscosity as follows:
Viscosity in gases arises from molecules traversing layers of flow and transferring momentum between layers. This transfer of momentum can be thought of as a frictional force between layers of flow. Since the momentum transfer is caused by free motion of gas molecules between collisions, increasing thermal agitation of the molecules results in a larger viscosity. Hence, gaseous viscosity increases with temperature.
In liquids, viscous forces are caused by molecules exerting attractive forces on each other across layers of flow. Increasing temperature results in a decrease in viscosity because a larger temperature means particles have greater thermal energy and are more easily able to overcome the attractive forces binding them together. An everyday example of this viscosity decrease is cooking oil moving more fluidly in a hot frying pan than in a cold one.
Thus for gases as the temperature rises the molecules move faster and thus transfer momentum (information about themselves) quicker whereas in a liquid the increased kinetic energy the molecules at a higher temperature, "vibrate" more, which results in it being easier for one layer of liquid to slip past an adjacent layer.