In the United States, there is a technical difference between BS(in nursing) and BSN just as their are differences in MSN and MS and between PhD and DSN (or DNSc, etc.) However, most people are not aware of it and it rarely makes a difference at the undergraduate or Master's levels.
The more specific BSN, MSN, DNS, etc. means that the degree is awarded from a professional school (of nursing) that determines the rules that govern the program. That school might be a designated school within a larger college/university, but the key fact is that it is recognized as a separate school in terms of some of the university's internal rules that govern how programs are run.
In PhD, MS, BS programs, the degree is award by the entire college or university -- in which the school of nursing is a "department," not a separate school. The total academic community usually has more influence in setting standards, etc. in such a program. At the undergraduate level, such things rarely make a difference. However, at the PhD level, it sometimes does. At the doctoral level, having your degree confirmed by the entire academic community (and NOT just the school of nursing) sometimes matters when seeking respect and acknowledgement from people in other disciplines. They want to know that the standards of the program were as high as they are for other disciplines. A DSN, etc. program can be just as good and maintain standards that are just as high, but if it has not gone through the same official approval process that other PhD programs have, there is not the official documentation of those standards and some people in other disciplines may not be so accepting of the degree -- particularly when it comes to evaluating academic standards.