Let the smaller number be \( a^3 \) and the larger number be \( b^3 \). According to the problem, we have:
\[ b^3 - a^3 = 631 \]
We also know that the cube root of the larger number is 1 more than the cube root of the smaller number. Thus, we can express this as:
\[ b = a + 1 \]
Substituting \( b \) in terms of \( a \) into the equation for the difference of cubes, we get:
\[ (a + 1)^3 - a^3 = 631 \]
Now, we can expand \( (a + 1)^3 \):
\[ (a + 1)^3 = a^3 + 3a^2 + 3a + 1 \]
Substituting this back into our equation, we have:
\[ a^3 + 3a^2 + 3a + 1 - a^3 = 631 \]
Simplifying this gives us:
\[ 3a^2 + 3a + 1 = 631 \]
Subtracting 631 from both sides, we obtain:
\[ 3a^2 + 3a + 1 - 631 = 0 \]
This simplifies to:
\[ 3a^2 + 3a - 630 = 0 \]
Now, we can divide the entire equation by 3:
\[ a^2 + a - 210 = 0 \]
Next, we need to find factors of \(-210\) that add up to \(1\). The factors of \(-210\) that fit this condition are \(15\) and \(-14\). Therefore, we can factor the quadratic:
\[ (a + 15)(a - 14) = 0 \]
Setting each factor to zero gives us:
\[ a + 15 = 0 \quad \Rightarrow \quad a = -15 \quad \text{(not a valid solution for cubes)} \]
\[ a - 14 = 0 \quad \Rightarrow \quad a = 14 \]
Now substituting \( a = 14 \) back to find \( b \):
\[ b = a + 1 = 14 + 1 = 15 \]
Thus the two numbers are:
\[ a^3 = 14^3 = 2744 \quad \text{and} \quad b^3 = 15^3 = 3375 \]
Finally, we verify the difference:
\[ 3375 - 2744 = 631 \]
Therefore, the two perfect cubes are \( 3375 \) and \( 2744 \).