If $\kappa$ is an infinite cardinal, then the relation $|X\bigtriangleup Y|\lt\kappa$ (where $X\bigtriangleup Y=(X\setminus Y)\cup(Y\setminus X)$) is an equivalence relation on $\mathcal P(\kappa)$. A collection $\mathcal E\subseteq\mathcal P(\kappa)$ is "diverse" if it contains at most one element of each equivalence class. $\mathcal E$ is a "maximal diverse family" if it contains exactly one element of each equivalence class, in which case $|\mathcal E|$ is equal to the number of equivalence classes, which is $2^\kappa$.

To see that the number of equivalence classes is $2^\kappa$, observe that $\{X\times\kappa:X\subseteq\kappa\}$ is a "diverse" collection of subsets of $\kappa\times\kappa$.

This argument does not apply to maximal almost disjoint families, because the relation $|X\cap Y|=\kappa$ is not an equivalence relation.