The flu and the common cold are caused by different viruses. These non-flu viruses include rhinovirus (one cause of the "common cold") and respiratory syncytial virus (RSV), which is the most common cause of severe respiratory illness in young children as well as a leading cause of death from respiratory illness in those aged 65 years and older. [CDC]
However, the flu and the cold are both respiratory illnesses.
Symptoms that come with the flu include high fever, body aches, extreme tiredness, and a dry cough. Often, these symptoms are more common and intense than they would be with the flu.
In the same vein, colds are usually milder than the flu and don't hit you as hard. Stuffy or runny noses are common with a cold.
Colds generally do not lead to serious medical conditions like pneumonia or bacterial infections, and do not usually require hospitalization.
Source: Center for Disase Control