as you know, there are varieties of adsorbents available (activated carbon, Chitosan, activated alumina, silica gel, molecular sieve carbon etc.). Most of them are manufactured (activated alumina) or processed from natural state (such as chitosan from chitin) and few of them occur naturally (say zeolites).
Each material has its unique characteristics such as porosity, pore structure and adsorption surfaces. Commercial adsorbents can be bought directly and they may have biological origin like chitosan or non-biological origin. But biosorbents are the low cost adsorbents (of course some exemptions like chitosan are there) and they have only the biological origin. Many materials of biological origin (e.g., fungi, yeast, bacteria, Chitosan, seeds of papaya, moringa oleifera and tamarind, peels of orange, banana and pomegranate and agricultural wastes) have been recognised as adsorbents for heavy metal ions.
You can study the difference between commercial adsorbents and biosorbents by conducting adsorption experiments by selecting one commercial adsorbent and one biosorbent on the sorptive removal of one particular metal ion. Comparative analysis can be done from that.