Trade Show, as name implies, is an event where companies in a specific industry gather to showcase and demonstrate their new products and services. In general Trade Shows are business-to-business (B2B) type of events; they are not open to the public and are meant to be attended by company representatives, industry officials and (of course!) press. Usually Trade Shows are organized and sponsored by trade associations for specific industries, so they are perfect networking platforms for industry professionals.
Trade Fair, on the other hand, is more business-to-client or customer (not only consumer!) type of event. Companies gather not only to showcase their products and services, but also to sell and market them. Trade fairs (or simply Fairs) are open to anyone interested and can cover a few industries at the same time. They tend to take place over 3-5 days, sometimes even longer.