We consider the set of four-digit positive integers $x =\overline{abcd}$ with digits different than zero and pairwise different. We also consider the integers $y = \overline{dcba}$ and we suppose that $x > y$. Find the greatest and the lowest value of the difference $x-y$, as well as the corresponding four-digit integers $x,y$ for which these values are obtained.
1. **Expression for \( x - y \)**:
   Given \( x = \overline{abcd} \) and \( y = \overline{dcba} \), we can express \( x \) and \( y \) in terms of their digits:
   \[
   x = 1000a + 100b + 10c + d
   \]
   \[
   y = 1000d + 100c + 10b + a
   \]
   Therefore, the difference \( x - y \) is:
   \[
   x - y = (1000a + 100b + 10c + d) - (1000d + 100c + 10b + a)
   \]
   Simplifying this, we get:
   \[
   x - y = 999a + 90b - 90c - 999d
   \]
   \[
   x - y = 999(a - d) + 90(b - c)
   \]

2. **Finding the Minimum Value**:
   To minimize \( x - y \), we need to minimize \( 999(a - d) + 90(b - c) \). Since \( a \) and \( d \) are digits from 1 to 9 and must be different, the minimum value of \( a - d \) is 1 (when \( a = d + 1 \)). Similarly, \( b \) and \( c \) are digits from 1 to 9 and must be different, so the minimum value of \( b - c \) is -8 (when \( b = c - 8 \)).

   Substituting these values:
   \[
   x - y = 999(1) + 90(-8) = 999 - 720 = 279
   \]

   This minimum difference is achieved when \( a = d + 1 \) and \( b = c - 8 \). For example, if \( a = 2 \), \( d = 1 \), \( b = 9 \), and \( c = 1 \), then:
   \[
   x = 2191 \quad \text{and} \quad y = 1912
   \]
   \[
   x - y = 2191 - 1912 = 279
   \]

3. **Finding the Maximum Value**:
   To maximize \( x - y \), we need to maximize \( 999(a - d) + 90(b - c) \). The maximum value of \( a - d \) is 8 (when \( a = 9 \) and \( d = 1 \)). Similarly, the maximum value of \( b - c \) is 8 (when \( b = 9 \) and \( c = 1 \)).

   Substituting these values:
   \[
   x - y = 999(8) + 90(8) = 7992 + 720 = 8712
   \]

   This maximum difference is achieved when \( a = 9 \), \( d = 1 \), \( b = 9 \), and \( c = 1 \). For example:
   \[
   x = 9812 \quad \text{and} \quad y = 2189
   \]
   \[
   x - y = 9812 - 2189 = 7623
   \]

The final answer is \( \boxed{279} \) for the minimum difference and \( \boxed{8712} \) for the maximum difference.