Neoprene is an industrial chemical used for rubber, coatings and insulation. Urethane can refer to two things: a class of chemicals known as polyurethanes, often used for adhesives or a carcinogen called ethyl carbamate.
Neoprene is used to synthesize rubber for wetsuits and sealants. Plugs of all kinds are made from neoprene, including those used to seal musical instruments and hoses. Neoprene is also used to make orthopedic braces and belts for car fans.
The polyurethane family of compounds is used to make adhesives, sealants and hard plastic components for electronics and appliances. Polyurethanes are not a single chemical: different compounds with certain similarities fall under the polyurethane category.
Polyurethane compounds are often toxic. Polyurethane glue, for example, exudes toxic fumes as long as it is wet. Fully-reacted polyurethane compounds are generally inert, but are still to be treated with caution. Ultraviolet rays can cause it to react. Neoprene, by contrast, is resistant to degradation by the sun and less reactive than polyurethanes.