Biogas is produced from decomposition of organic matter, usually in a digester of some sort. It is a literal gas, primarily methane and
CO2. Think propane substitute.
Biodiesel is refined via chemical reaction from veg oil; methanol+lye are combined to make 'methoxide', which is mixed with the veg oil in a reactor; this then separates into biodiesel, glycerin, and some other minor byproducts.
It is a liquid fuel, compatible with pretty well all applications where diesel is used.
Too long, but that's what I've got right now..