We present a new multicomponent SED decomposition method and use it to analyze the UV to millimeter wavelength SEDs of a sample of dusty infrared-luminous galaxies. Each SED is decomposed into emission from populations of stars, an AGN accretion disk, PAHs, atomic and molecular lines, and distributions of graphite and silicate grains. Decompositions of the SEDs of template starburst galaxies and AGNs provide baseline properties to aid in quantifying the strength of star formation and accretion in the composite systems NGC 6240 and Mrk 1014. We find that obscured radiation from stars is capable of powering the total dust emission from NGC 6240. The presence of a small quantity of 1260 K dust in this source suggests a ~2% AGN contribution, although we cannot rule out a larger contribution from a deeply embedded AGN visible only in X-rays. The decomposition of Mrk 1014 is consistent with ~65% of its power emerging from an AGN and ~35% from star formation. We suggest that many of the variations in our template starburst SEDs may be explained in terms of the different mean optical depths through the clouds of dust surrounding the young stars within each galaxy. Prompted by the divergent far-IR properties of our template AGNs, we suggest that variations in the relative orientation of their AGN accretion disks with respect to the disks of the galaxies hosting them may result in different amounts of AGN-heated cold dust emission emerging from their host galaxies. We estimate that 30%-50% of the far-IR and PAH emission from Mrk 1014 may originate from such AGN-heated material in its host galaxy disk.