  Given an ACM set $\mathbb{X}$ of points in a multiprojective space
$\mathbb{P}^m\times\mathbb{P}^n$ over a field of characteristic zero, we are
interested in studying the K\"ahler different and the Cayley-Bacharach property
for $\mathbb{X}$. In $\mathbb{P}^1\times\mathbb{P}^1$, the Cayley-Bacharach
property agrees with the complete intersection property and it is characterized
by using the K\"ahler different. However, this result fails to hold in
$\mathbb{P}^m\times\mathbb{P}^n$ for $n>1$ or $m>1$. In this paper we start an
investigation of the K\"ahler different and its Hilbert function and then prove
that $\mathbb{X}$ is a complete intersection of type
$(d_1,...,d_m,d'_1,...,d'_n)$ if and only if it has the Cayley-Bachrach
property and the K\"ahler different is non-zero at a certain degree. When
$\mathbb{X}$ has the $(\star)$-property, we characterize the Cayley-Bacharach
property of $\mathbb{X}$ in terms of its components under the canonical
projections.
