Biological has always had a notable edge, but it seems subjectively to have eroded slowly over time over the last 40+ years that I’ve been exposed to it (thinking mostly of print, its been near parity in what I’ve heard, but “birth” is what I favor and what my [adoptive] parents favored so…)
The birth parents would be the mother who actually gave birth (and if existing, the spouse who is in the relationship with the birth mother).
This is would be much more prevalent of a distinction due to surrogacy.
Also, I'd assume the child would later look for those people with their genetic material, their biological parents, those who provided the sperm & eggs. Not the birth parent, who birthed them. There's much less of a connection there as it's the person who simply carried them for a money transaction and is not part of the child genetically.