We characterize those measures μ for which the Hardy-Orlicz (resp., weighted Bergman-Orlicz) space HΨ1 (resp., AαΨ1) of the unit ball of CN embeds boundedly or compactly into the Orlicz space LΨ2(BN¯,μ) (resp., LΨ2(BN,μ)), when the defining functions Ψ1 and Ψ2 are growth functions such that L1⊂LΨj for j∈{1,2}, and such that Ψ2/Ψ1 is nondecreasing. We apply our result to the characterization of the boundedness and compactness of composition operators from HΨ1 (resp., AαΨ1) into HΨ2 (resp., AαΨ2).