An allergy is an immune system response. It is caused when the body mistakes a protein as a harmful substance and creates a defence system (antibodies) to fight it.
Allergic reactions are immediate reactions, and they can be life-threatening in some cases.
An intolerance is a non-immune system response. Symptoms of intolerance usually occur several hours after exposure to a trigger, whereas an allergy can result in an immediate reaction.
Often people need to be exposed to larger amounts of food types to trigger an intolerance unlike an allergy. A food intolerance is not life-threatening.