There is no difference. There is no global standard for what qualifies as a BS and what qualifies as a BA; it's entirely up to the university. There is no way to tell from the letters on your diploma how broad or rigorous your degree program is.
Computer science programs in the US, at least in engineering colleges, are accredited by ABET. Those accredited programs tend to give Bachelor of Science degrees. But CS accreditation is relatively new, and neither graduate scihools nor employers really care whether your degree is accredited (except for some government jobs which require accredited degrees by law).
What does matter is which classes you took, and how rigorous those classes are (or appear to be, based on publicly available course descriptions/materials).
If the degree type doesn't matter, what are the positives/negatives of getting a PhD with an BA/BS.
The only advantage of getting a PhD with a BS instead of a BA is that you have one more S in your CV and one less A. This is also the only disadvantqge.