We report the results from a comprehensive study of 74 ultraluminous infrared galaxies (ULIRGs) and 34 Palomar-Green (PG) quasars within z ~ 0.3$ observed with the Spitzer Infrared Spectrograph (IRS). The contribution of nuclear activity to the bolometric luminosity in these systems is quantified using six independent methods that span a range in wavelength and give consistent results within ~ +/-10-15% on average. The average derived AGN contribution in ULIRGs is ~35-40%, ranging from ~15-35% among cool (f_25/f_60 =< 0.2) optically classified HII-like and LINER ULIRGs to ~50 and ~75% among warm Seyfert 2 and Seyfert 1 ULIRGs, respectively. This number exceeds ~80% in PG QSOs. ULIRGs fall in one of three distinct AGN classes: (1) objects with small extinctions and large PAH equivalent widths are highly starburst-dominated; (2) systems with large extinctions and modest PAH equivalent widths have larger AGN contributions, but still tend to be starburst-dominated; and (3) ULIRGs with both small extinctions and small PAH equivalent widths host AGN that are at least as powerful as the starbursts. The AGN contributions in class 2 ULIRGs are more uncertain than in the other objects, and we cannot formally rule out the possibility that these objects represent a physically distinct type of ULIRGs. A morphological trend is seen along the sequence (1)-(2)-(3), in general agreement with the standard ULIRG - QSO evolution scenario and suggestive of a broad peak in extinction during the intermediate stages of merger evolution. However, the scatter in this sequence, implies that black hole accretion, in addition to depending on the merger phase, also has a strong chaotic/random component, as in local AGN. (abridged)
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