What is the difference between a chemical engineer and a nuclear engineer?

The main difference is specilization. I am a chemical engineer myself, and we have the option of specializing in a wide variety of degrees, such as genetic engineering. Note that chemical engineering is a very broad field dealing with polymers, Metallurgy, Biology, environmental chemistry, and the list goes on. Genetic engineering, in short, is really a specialization of chemical engineering to specifically biological chemistry.