The primary difference across the schools is the level of resources that the school has.
Smaller high schools many not have many APs or activities compared to richer public or private schools.
At the very top, however, things are very different.
Admissions departments at the top schools get to know the guidance counselors at the top schools, both public and private, and over the years, are able to tell which are the top students because they know what level of academics and activities come from the school.
Also, because of the relationships with the guidance counselors, if they receive a strong recommendation from a guidance counselor that they know, it will be more helpful than from someone they have no relationship with.
Top private schools such as Philips Exeter, Andover, Deerfield regularly send many students to the Ivy League each year.
Similarly, top public schools such as Thomas Jefferson High School for Science and Technology, and Stuyvesant will send equally many students to the Ivy League each year.