Does air resistance affect the motion of a falling object differently when the initial velocity of the object is greater?

Yes, it does.

We have that air resistance creates a force on falling objects, namely drag. The drag of an object depends on its shape, its cross-section in the direction it is falling, the nature of the air (humind or not) and the velocity of the object. In general, the greater the velocity, the greater the drag; the drag force is against the motion of the body.

Supplementary info: 1)It is also calculated that raindrops and other small objects, when they come down to Earth they develop very high velocities. If we account for drag, the force of drag balances out the force of gravity due to their great velocity and they stop accelerating, they have reached terminal velocity.

2)This is contrary to our experience with friction, that does not depend on velocity almost at all.