Biosynthesis means using an extract of plants or microbes. Green synthesis is generally the just to minimize the effect of toxic chemicals. If you are using water as solvent instead of toxic organic solvent then that process will be green.
According to my knowledge, synthesis procedure that do not emit any green house gases and hence considered as clean and environmental friendly can be called as green synthesis whereas, the low cost or economical synthesis that involves biological wastes as a raw material can be considered as bio synthesis.
Green synthesis just involve plants while biosynthesis involve synthesis from Fungi, Algae, Microbes and plants. In 2017 paper has been published, where nanoparticles synthesized from DNA, it also comes in category of Biosynthesis. Synthesis from microbes are complicated process while synthesis from Plants are easy. Because plants are available everywhere and rich source of metabolites.
Biosynthesis means using an extract of plants or microbes. Green synthesis is generally the just to minimize the effect of toxic chemicals. If you are using water as solvent instead of toxic organic solvent then that process will be green.