Both the flu and the cold are caused by viruses and neither one is less contagious than the other virus.
Both the cold and flu virus can be contagious for as long as 7 days so if you or your child has the flu or a cold you should stay home and the child shouldn't go back to school or the adult shouldn't go back to work until at least 10 days have passed to keep from spreading the flu or cold virus.
Colds are an upper respiratory infection and the flu is considered a lower respiratory infection.