The student throws one cannonball directly upward at 5.0 m/s and simultaneously throws the other cannonball directly downward at 5.0 m/s. (this requires great dexterity, but the student has been practicing this maneuver for several months.) what is the difference in speed between the cannonballs when each ball strikes the ground?

Let the cannonball be thrown at a height of h above ground.
Then  the potential energy of the ball is
V = m*g*h
where
m = the mass of the ball
g = 9.8 m/s²

Also, the kinetic energy of the ball is
K = (1/2)mu²
where
u = 5 m/s, the vertical launch velocity.
Ignore wind resistance.

Because the total energy is preserved, the total energy (n the form of only kinetic energy) when the ball strikes the ground is
(1/2)mV²
where V = vertical velocity when the ball strikes the ground.

Expressions for both the initial and final energy are equal regardless of whether the ball s thrown downward or upward.
Therefore there is no difference in the landing speed. 

There is no difference.