The damping coefficients of steel and even aluminum both are low enough so as to be almost insignificant as compared to the tires, saddle and rider's arse that exist in the load path between the road and the rider's mass.
My reasoning here is that tires and body tissues can damp out a vibratory impulse within a few cycles, whereas most metals won't damp out a vibratory impulse even within 20 cycles because of the low numbers associated with their damping coefficient.

Where different metals could be more noticeably different is in how they damp out lateral vibrations that are at 90-degrees to the tube axis, where a vibration that is perpendicular to the tube and thus load path might go on vibrating for many cycles. In this case, the sound and possibly the buzziness of high-frequency felt vibrations could be much lower for aluminum with it's higher damping coefficient relative to steel. I have no idea though whether a rider can actually feel the vibrations of a tube bowing in such an unloaded cyclic fashion, it's frequency depending on it's mass and stiffness. The higher vibration frequency of a large-diameter tube would tend to be more quickly attenuated by the metal itself, but likely even more by the tires and rider attached to the frame structure.