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Molecular dynamics simulations and lattice dynamics calculations are used to study the vibrational modes and thermal transport in Lennard-Jones superlattices with perfect and mixed interfaces. The secondary periodicity of the superlattices leads to a vibrational spectrum (i.e., dispersion relation) that is distinct from the bulk spectra of the constituent materials. The mode eigenvectors of the perfect superlattices are found to be good representations of the majority of the modes in the mixed superlattices for up to 20% interfacial mixing, allowing for extraction of phonon frequencies and lifetimes. Using the frequencies and lifetimes, the in-plane and cross-plane thermal conductivities are predicted using a solution of the Boltzmann transport equation (BTE), with agreement found with predictions from the Green-Kubo method for the perfect superlattices. For the mixed superlattices, the Green-Kubo and BTE predictions agree for the cross-plane direction, where thermal conductivity is dominated by low-frequency modes whose eigenvectors are not affected by the mixing. For the in-plane direction, mid-frequency modes that contribute to thermal transport are disrupted by the mixing, leading to an underprediction of thermal conductivity by the BTE. The results highlight the importance of using a dispersion relation that includes the secondary periodicity when predicting phonon properties in perfect superlattices and emphasize the challenges of estimating the effects of disorder on phonon properties. [ABSTRACT FROM AUTHOR]