The movement of the magnet induces an electric current in the copper and with electric current comes a magnetic field, which attracts the magnet. The magnet doesn't stick to the wall as it falls because the induced current, and its corresponding magnetic field, are perfectly distributed so that the magnet feels magnetic force equally from all sides. The magnetic field slows the magnet, but can't stop its fall because if the magnet stopped moving, the induced electric field would go away and the magnet would start falling again.
Of note, while the magnet is falling slowly, the copper pipe will feel heavier in the hand because the pipe is "holding up" the magnet. I wonder whether the same effect could be observed by dropping buckyballs through a smaller copper tube?