  A function $F:\mathbb{F}_{p^n}\rightarrow \mathbb{F}_{p^m},$ is a vectorial
$s$-plateaued function if for each component function $F_{b}(\mu)=Tr_n(\alpha
F(x)), b\in \mathbb{F}_{p^m}^*$ and $\mu \in \mathbb{F}_{p^n}$, the Walsh
transform value $|\widehat{F_{b}}(\mu)|$ is either $0$ or $ p^{\frac{n+s}{2}}$.
In this paper, we explore the relation between (vectorial) $s$-plateaued
functions and partial geometric difference sets. Moreover, we establish the
link between three-valued cross-correlation of $p$-ary sequences and vectorial
$s$-plateaued functions. Using this link, we provide a partition of
$\mathbb{F}_{3^n}$ into partial geometric difference sets. Conversely, using a
partition of $\mathbb{F}_{3^n}$ into partial geometric difference sets, we
constructed ternary plateaued functions $f:\mathbb{F}_{3^n}\rightarrow
\mathbb{F}_3$. We also give a characterization of $p$-ary plateaued functions
in terms of special matrices which enables us to give the link between such
functions and second-order derivatives using a different approach.
