CDC states the following:

• “Influenza (flu) and the common cold are both contagious respiratory illnesses, but they are caused by different viruses.
• Colds and flu share many symptoms, so it can be difficult to tell the difference between them based on symptoms alone.
• Cold symptoms are usually milder than flu.
• Special tests can tell if a person is sick with flu.”

Centers for Disease Control and Prevention (https://www.cdc.gov/flu/about/coldflu.html)