Let's state that a clock has 60 degrees, whereby each degree on a clock is 6 degrees on a circle (for a total of \( 60 * 6 = 360 \) degrees).

The angle at 4:00 has the minute-hand on the 12, and the hour-hand on the 4. This means that the smallest angle is \( 4 * 5 * 6 = 120 \) degrees.

The angle at 4:15 has the minute-hand on the 3, and the hour-hand one clock degree away from the 4. The smallest angle is given by \( 6 * 6 = 36 \) degrees.

Therefore, the difference in angles is \( 120 - 36 = 84 \) degrees.