A plastic ball and a metal ball of similar shape and size are dropped. The metal ball has a greater mass. Why does the plastic ball reach a constant velocity but the metal ball does not? And why does both the balls have the same initial acceleration?

As a metal ball of greater mass is dropped at the same time as a plastic ball of less mass, the only acceleration that initially acts on both balls is the acceleration caused by gravitational force of the Earth on both balls; this acceleration, g, is a constant of approximately 9.81 meters per second squared. In "free fall" where there is no air resistance, both balls would land on the ground at the exact same time because the only acceleration acting on both balls would be g, ensuring that both balls would have the same velocity at any given point in time during their fall.
However, when air resistance is considered, it's a whole different story. When an object falls, it gets faster and faster, and as it goes down, it encounters the upward force of air resistance. This air resistance is caused by the object falling and pushing against the air itself on its way down, and colliding with air molecules which push back up on the object falling. As the object gains speed, it encounters a greater air resistance force.
As both balls fall, the ball of greater mass experiences a greater downward force of gravity, as Force = mass * acceleration (F=ma). As both balls have the same acceleration g, the ball with more mass, the metal one, will have a greater force F. Now, as both balls fall, they experience the same upward force of air resistance. Yet because the plastic ball has a smaller downward gravitational force, eventually the upward air resistance force will equal the downward gravitational force on the ball. These two forces will cancel out, resulting in zero net external force on the plastic ball, causing it to stop accelerating and reach a constant velocity, also called the terminal velocity. Meanwhile, the metal ball of greater mass has a greater downward force, and as it falls, the air resistance force never gets large enough to equal the downward force, hence there is always a net force downward on the metal ball, causing it to keep accelerating and not reach a constant terminal velocity.
http://hyperphysics.phy-astr.gsu.edu/hbase/airfri2.html