What's the difference between them? Do mostly Recruited Athletes go to NAPS and others go to Foundation? Which one do I, if I don't get appointed, have a better chance to be offered, granted academics is the reason for being denied to USNA?
The U.S. Naval Academy Foundation helps "highly qualified candidates by providing scholarship opportunities to enable aspiring young men and women to attend private preparatory schools."
(Pays to get you in a prep school, which will better your chances of getting offered an appointment in the next class.)
The Naval Academy Preparatory School (NAPS) mission is to "enhance midshipman candidates' moral, mental, and physical foundations to prepare them for success at the U.S. Naval Academy."
(Is a prep school. The 10-month program there is designed to get you ready to join the next year's USNA class. If you go and do well, you will more than likely be offered an appointment.)
NAPSters enlist for a year, start NAPS in late July/early August, and are paid while there. NAPS is free to those who attend, is primarily taxpayer supported, and is a military school. It's a bit more of a "mini" plebe year in that you learn certain things that will help a lot plebe year, such as how to fold your clothes, how to wear a uniform, how to march, ranks and other military info, etc.
Foundation schools are civilian schools, though some are "military" prep schools in that they have a military environment. Most are high schools that offer an additional "prep" year, though a handful offer college credit. Foundation schools are supported by USNA Foundation scholarships (private funding) and parents are expected to contribute as well, based on their ability to pay.
Students at both types of schools are required to take the plebe courses, do well athletically, and keep their noses clean. If they successfully complete their programs and still want to attend USNA, they are virtually guaranteed an appointment.
Recruited athletes and others (those who are not recruited athletes) attend both NAPS and Foundation schools. I don't believe there is any formal delineation in terms of where certain recruited athletes go. As a practical matter, those recruited for football, basketball (men's and women's) and men's lacrosse who are sent to prep school typically end up at NAPS. Recruited athletes from other sports who are sent to prep shcool often end up at Foundation schools. The prior is, however, a generalization -- there are undoubtedly exceptions. Also, many recruited athletes (in all sports) are direct appointees and do not go to any prep school.