We compare the compactness of composition operators on and on Orlicz–Hardy spaces . We show that, for every , there exists an Orlicz function Ψ such that for every , and a composition operator which is compact on and on , but not on . We also show that, for every Orlicz function Ψ which does not satisfy condition , there is a composition operator which is compact on but not on , and that, when Ψ grows fast enough, there is a function ϕ such that is in all Schatten classes , for , but is not compact on .