We consider the decimal representation of the integers
$$\begin{aligned} x - y &= 1000\alpha + 100\beta + 10\gamma + \delta - 1000\delta - 100\gamma - 10\beta - \alpha \\ &= 999(\alpha - \delta) + 90(\beta - \gamma) = 9(111(\alpha - \delta) + 10(\beta - \gamma)). \end{aligned}$$
Therefore it is enough to find the greatest and the lowest values of the expression:
$$A = 111(\alpha - \delta) + 10(\beta - \gamma).$$
Since $\alpha, \beta, \gamma, \delta$ are digits pairwise different and no zero, we must have $\alpha > \delta$. The expression $A$ gets its maximal value when the integers $\alpha - \delta$ and $\beta - \gamma$ take their maximal value and moreover $\alpha - \delta > \beta - \gamma$. That is, $\alpha - \delta$ becomes maximal when $\alpha = 9$ and $\delta = 1$. The difference $\beta - \gamma$ becomes maximal when $\beta = 8$ and $\gamma = 2$. Hence $x = 9821$, $y = 1289$ and $x - y = 9821 - 1289 = 8532$.
The expression $A$ gets its minimal value when the integers $\alpha - \delta$ and $\beta - \gamma$ take their minimal values. The minimal value of the difference $\alpha - \delta$ is $1$ and so the pair $(\alpha, \delta)$ has as possible values:
$$(9,8), (8,7), (7,6), (6,5), (5,4), (4,3), (3,2) \text{ and } (2,1).$$
For all the above possible values of the pair $(\alpha, \delta)$, the value of $A$ is:
$$A = 111 + 10(\beta - \gamma).$$
The lowest value of the difference $\beta - \gamma$ is $-8$, for $\beta = 1$ and $\gamma = 9$.
Taking in mind that the digits must be different we find the following table of possible values:

| $x$ | $y$ | $x-y$ |
|-------|-------|-------|
| 3192 | 2913 | 279 |
| 4193 | 3914 | 279 |
| 5194 | 4915 | 279 |
| 6195 | 5916 | 279 |
| 7196 | 6917 | 279 |
| 8197 | 7918 | 279 |