BSN vs. BS with nursing major?

For practical purposes, there is no difference. The title of the degree is determined by the school -- and is usually based on the internal organization of the institution and how their administration is organized.

If the school is organized so that the nursing department is a separate "college" within the university system, then they usually award a BSN. If the nursing department is not a separate school with it's own dean and separate administration, then they usually call their degrees a BS (or BA) with a major in nursing. Of course, you will probably find a few exceptions to that based on local tradition and culture. A few schools offer both degrees with slightly different requirements -- with the BS focusing more on academics and the BSN focusing more on practice -- but it is very uncommon for a school to offer both. Most offer one or the other and the requiements for either one vary more from school to school than between degree titles.

If the schools are reputable and accredited, it makes no difference.

llg, PhD, RN