Somee University's have found that they actually had no policy to differentiate why some degrees were only offered as a BA or as BS and decided to create (or at least formally define) a difference, like this story:
BA/BS, Difference Between Degrees
Background: in 1984, the Academic Senate Curriculum Committee was
asked to prepare a policy statement which would differentiate between
Bachelor of Science degrees and Bachelor of Arts degrees at California
Polytechnic State University, San Luis Obispo. The need for this arose
when some departments at Cal Poly proposed to change their B.A.
degrees to B.S. degrees. The requests were refused by the Chancellor's
Office in part because Cal Poly, SLO, had no campus policy which
specified the differences between the two degrees.
This one University created this type of distinction:
Bachelor of Arts Degree:
- is usually less specialized than a Bachelor of Science degree.
- requires a minimum of 180 quarter units for the degree; 36 units are
required in the major, of which at least 18 units are at the 300-400
- is normally awarded in such majors as the languages,
literature, other humanities, and history.
Bachelor of Science Degree:
- typically involves technical fields.
- requires a minimum of 180 quarter
units for the degree; 54 units are required in the major, of which at
least 27 quarter units are at the of 300-400 level.
- is normally
awarded in such majors as the physical and biological sciences,
engineering, and agriculture
Here's a different take on the same issue:
The BA is a liberal arts degree intended to develop skills and talents
essential to succeeding in a global society. It combines study of the
arts, humanities, historical perspectives, and the natural and social
sciences with advanced critical inquiry and an in-depth knowledge in
an academic discipline (major). The program develops cultural, social,
and political literacy, including the abilities to communicate
effectively and clearly in writing and in speech, and to understand on
a basic level a world language other than one’s own.* In so doing, it
fosters the ability to understand and actively participate in
discourses both within and beyond the field of your major (such as a
minor or second major), and it promotes engagement with cultural,
social, and political difference.
The BS is a liberal arts degree intended to develop skills and talents
essential to professional work. It combines study of the arts,
humanities, historical perspectives, and the natural and social
sciences with advanced critical inquiry and an in-depth knowledge in a
specific academic discipline (major). The program develops an
understanding of empirical analysis, scientific methodology and
protocols, and mathematics and quantitative techniques. In so doing,
it equips you for continued engagement in professional research within
your chosen field.
What's the actual difference in terms of coursework, other than the choice of major itself? The BA requires a foreign language (2 semesters) and a few extra humanities classes, while the BS has no requirement for foreign language but adds an extra natural science lab and extra math class. It's mostly a nonsense differentiation that means nothing, but it sounds fancy and gives the illusion of choice if you don't know why you want to go to college - so it's an effective part of the marketing to some audiences.
But in all of these note that no mention is made of the scientific method, math, or any classically recognized type of 'art' (such as painting, literature, etc). The words art and science are not well defined in this way across academia, and so in the end they are used in whatever way the school decides they want to use them.
There are some other classifications in use as well, such as bachelors of: music (BM), applied science (BAS), fine art (BFA), and even a Bachelor Of Business Administration (BBA).
One final note is the classification sometimes ends up purely as a result of what 'school' or 'college' within the University grants the degree. Some schools define themselves as liberal arts/humanities schools and only offer a BA, some define themselves as technical/technology schools and only grant a BS, business schools are more likely to offer something branded as a BBA/MBA, schools of "letters and science" often offer both, etc. In such schools the difference is who is in charge of that particular degree, more than it reflects a genuine difference in what the classification means to students or other community members.
And all of this is just in the US system!
TLDR; The difference in a BA/BS and how "business administration" will be classified (as BA, BS, BBA, etc) means something only within the context of a given school, varies widely between schools, and sometimes means nothing. Each institution generally chooses for itself what, if any, difference there is, so you'll need to refer to each school for reasoning on why they do things they way they do - and sometimes the answer will just be tradition. 'Art' and 'science' in academic degrees don't necessarily mean the same thing as they do in the vernacular.