A biofilter is simply a filter media that removes something specific out of water, like Phosphorus for example. An example of a biofilter would be a bag of a granulated Phosphate removal product. Most are iron oxide pellets.

A microfilter filters out certain types of microorganisms, like bacteria. Water is forced through a specific kind of membrane. Better models remove all but viruses. An Ultraviolet Sterilizer can remove more microorganisms than a microfilter, but there are portable microfilters. I have one in my camping supplies to make stream water drinkable.