This paper is best described by its abstract: “We construct an analytic self-map \( \varphi \) of the unit disk and an Orlicz function \( \Psi \) for which the composition operator of symbol \( \varphi \) is compact on the Hardy-Orlicz space \( H^\Psi \), but not 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 \( 2\).”