  There are two construction methods of designs from $(n,m)$-bent functions,
known as translation and addition designs. In this paper we analyze, which
equivalence relation for Boolean bent functions, i.e. $(n,1)$-bent functions,
and vectorial bent functions, i.e. $(n,m)$-bent functions with $2\le m\le n/2$,
is coarser: extended-affine equivalence or isomorphism of associated
translation and addition designs. First, we observe that similar to the Boolean
bent functions, extended-affine equivalence of vectorial $(n,m)$-bent functions
and isomorphism of addition designs are the same concepts for all even $n$ and
$m\le n/2$. Further, we show that extended-affine inequivalent Boolean bent
functions in $n$ variables, whose translation designs are isomorphic, exist for
all $n\ge6$. This implies, that isomorphism of translation designs for Boolean
bent functions is a coarser equivalence relation than extended-affine
equivalence. However, we do not observe the same phenomenon for vectorial bent
functions in a small number of variables. We classify and enumerate all
vectorial bent functions in six variables and show, that in contrast to the
Boolean case, one cannot exhibit isomorphic translation designs from
extended-affine inequivalent vectorial $(6,m)$-bent functions with $m\in\{ 2,3
\}$.
