Biological weapons - as the name suggests, use a biological agent - usually a virus or bacteria or spore which has been engineered specifically for persistance, virulence and genral nastiness - stuff like Smallpox, Anthrax, Spanish Flu and the like
A chemical weapon is a pure chemical, some are developed specificall for warfare but some are discovered by chance or as an offshoot of - for exmaple - pesticide development. They are mostly designed to penetrate the skin or be inhaled, causing internal and external damage - examples are Phosgene, Mustard Gas, VX, Tabun
In order to maximise effectiveness of delivery they are usually delivered by airborne weapon suck as a rocket which detonates in the atmosphere creating a cloud of nastiness. They can also be use don a smaller scal as was seen in the USA with the "Anthrax in an envelope" terror attacks.
As to which one is nastier, hard to say. Biological weapons can infect a population which could in theory help to perpetuate the infection and spread it much further wheras chemical agents cannot reproduce. Chemical weapons are more persistent - contamination in the soil and on surfaces can last for many years (although the same can be said for Anthrax). Chemical weapons have a more immediate effect as opposed to the incubation period associated with biological contamination. You can vaccinate against some bio-weapons but only protection against chemical weapons is barrier protection (gas masks, NBC Suits). Both are nasty and indescriminate weapons - hence their use is not condoned.