APACHE II was designed to provide a morbidityscore for a patient. It is useful to decide what kind of treatment or medicine is given. Methods exist to derive a predicted mortality from this score, but these methods are not too well defined and rather imprecise.
SAPS II was designed to provide a predicted mortality, that does not reflect the expected mortality for a particular patient, but is good for benchmarking. In a rather simple way, it makes it possible to provide a single number that describes the morbidity of a number of patients.
SAPS III was designed to provide a realistic predicted mortality for a particular patient or a particular group of patients. It does this by calibrating against known mortalities on an existing set of patients, for a specific definition of mortality (like 30-days mortality). This way, it can answer questions like "Did we improve our quality of care from 2004 to 2005?" or "If hospital A's patients had been treated at hospital B, would they have a better or a worse mortality?".
^ abSoares M, Fontes F, Dantas J, Gadelha D, Cariello P, Nardes F et al. (2004). "Performance of six severity-of-illness scores in cancer patients requiring admission to the intensive care unit: a prospective observational study.". Crit Care8 (4): R194–203. doi:10.1186/cc2870. PMC522839. PMID15312218.