A $2 \times 4 \times 8$ rectangular prism and a cube have the same volume. What is the difference between their surface areas?
1. **Calculate the volume of the rectangular prism:**
   The volume \( V \) of a rectangular prism is given by the product of its length, width, and height.
   \[
   V = 2 \times 4 \times 8 = 64
   \]

2. **Determine the side length of the cube:**
   Since the cube has the same volume as the rectangular prism, we set the volume of the cube equal to 64.
   Let \( s \) be the side length of the cube. Then,
   \[
   s^3 = 64
   \]
   Solving for \( s \),
   \[
   s = \sqrt[3]{64} = 4
   \]

3. **Calculate the surface area of the rectangular prism:**
   The surface area \( A \) of a rectangular prism is given by:
   \[
   A = 2(lw + lh + wh)
   \]
   Substituting \( l = 2 \), \( w = 4 \), and \( h = 8 \),
   \[
   A = 2(2 \times 4 + 2 \times 8 + 4 \times 8) = 2(8 + 16 + 32) = 2 \times 56 = 112
   \]

4. **Calculate the surface area of the cube:**
   The surface area \( A \) of a cube with side length \( s \) is given by:
   \[
   A = 6s^2
   \]
   Substituting \( s = 4 \),
   \[
   A = 6 \times 4^2 = 6 \times 16 = 96
   \]

5. **Find the difference between the surface areas:**
   \[
   \text{Difference} = 112 - 96 = 16
   \]

The final answer is \(\boxed{16}\).