The SAT is a standardized test. Your test score is related to your answers on the test.
The "adversity score" is a 'further information' about the student that takes the 15 factors (whatever they are) and creates a score on those factors (presumably with handy graphs). It is provided to institutions to give an impression of the student beyond the test score. Institutions can use it or not.
Two things to note within this, one is that most schools are looking at multiple factors for enrollment now. Take SAT, take HS GPA, take rigor of coursework taken, and take AP, IB or dual enrollment credit into consideration (and for highly selective schools, volunteer work, references, etc). Theoretically this information - standardized by the SAT - could also be used in a multi-factor decision process. Its entirely possible that some schools may have a local version already concocted from their own data about student success at their school.
The second is that current data shows that while SAT is a good measure of retention of information, retention of information is not always a good indicator of success at school. For a traditional freshman, current data suggests that HS GPA (even considering grade inflation) is actually a better indicator of potential success in college (but even then, not perfect, hence the use of multiple factors).