The key physics in play here is twofold. Absent other forces, the two objects undergo the exact same acceleration in free fall, and so will fall at the same rate. With no air in the tube, the only force acting on them is gravity, which pulls downward on each object proportional to its mass.
However, when air is in the tube, there is a second force involved: air resistance.
The force of air resistance pushes upwards on the falling objects. It depends on two factors: the surface area of the falling object, and its velocity. So the faster they fall, the more resistance they face from the air. But recall that the force of gravity is proportional to the mass of the object, and the net acceleration of an object is the result of the sum of the forces acting on it. So if two objects have similar surface area, but one has a higher mass, then the higher mass one experiences a larger downward force than the other, while air resistance will exert close to the same upward force on both, and so the heavier object then has a greater acceleration. And that’s what we see when the tube is full of air – the more massive disc falls faster than the less massive feather. Take away the air and the force of air resistance, and they fall together!Also see demonstration C4-33 a similar but larger demonstration for use in lecture halls.