Donna said, "I think a falling Apple would accelerate more slowly on the moon than on earth because the force of gravity is less." Anita said, "I...

They are both right.

Donna is right that the force of gravity is less on the moon, so by Newton's Second law a falling apple will obtain less acceleration on the moon—that is, if we ignore air resistance on Earth, which is generally incorrect, as Anita said.

Note that air resistance for a fixed object depends on its speed. So when an apple just starts falling, its speed is almost zero, and air resistance is almost zero, too. Therefore soon after its starts moving, an apple on Earth will have greater acceleration.

When both apples will gain considerable speed, the acceleration of an apple on Earth will become less than at the start, and will tend to zero (the speed will reach its maximum). The acceleration of an apple over the moon will be constant and the speed may be very high.

In between, there will be a time moment when the accelerations will become equal.

This way, Donna's reasoning is better for small speeds, while Anita's better for high speeds.