We construct an analytic self-map $\phi$ of the unit disk and an Orlicz
function $\Psi$ for which the composition operator of symbol $\phi$ is compact
on the Hardy-Orlicz space $H^\Psi$, but not compact on the Bergman-Orlicz space
${\mathfrak B}^\Psi$. For that, we first prove a Carleson embedding theorem,
and then characterize the compactness of composition operators on
Bergman-Orlicz spaces, in terms of Carleson function (of order 2). We show that
this Carleson function is equivalent to the Nevanlinna counting function of
order 2.