Nuclear power plants and coal-fired power plants share many similarities in terms of their function and operation; there are, however, several key differences. The most important of these lies in the nature of the reaction they use for energy.
Both coal-fired and nuclear power plants are thermal power plants, meaning that they use heat energy to vaporize water and make superheated steam. The steam then drives a turbine, which turns a generator to generate electricity. The differences lie in the nature of the process used to heat the water.
Nuclear power plants produce heat using nuclear reactions, which convert mass into energy by splitting the nuclei of atoms of heavy elements like uranium. Coal-fired power plants, on the other hand, use chemical reactions (the combustion of coal) to produce heat. In a chemical reaction, mass is conserved; only the bonds between atoms in molecules are changed.
The nuclear reactions in a nuclear power plant create unstable isotopes as byproducts of the reaction; these unstable isotopes are highly radioactive and hence must be buried until they have decomposed to where the radiation is at a safe level. Coal-fired power plants, on the other hand, release pollutants in the form of sulfur dioxides, carbon monoxide/dioxide and other compounds formed during combustion.