  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)
