A food allergy is a sensitivity caused by a reaction of the body's immune system to specific proteins in a food. While food intolerances are food sensitivities that do not involve the individual's immune system.

A food allergy is a sensitivity caused by a reaction of the body's immune system to specific proteins in a food. Food allergy symptoms occur when the immune system decides that a protein in a food is harmful, and begins an attack when that food enters the body.
The most common food allergies are peanuts, tree nuts (such as almonds and walnuts), sesame seeds, milk, eggs, seafood (fish, crustaceans, and shellfish), soy, wheat, and sulfites (a food additive).
The Canadian Food Inspection Agency works with the food industry to ensure that the common food allergies are always declared in the ingredient list on the food label and that they develop allergen prevention strategies to manage the allergy risk.
The most common symptoms of a food allergy are:
Skin problems including hives; tissue swelling, especially of the face and around the mouth; itching and reddening of the skin; and eczema in young children.
Breathing problems; stuffy, runny nose and itchy eyes; worsening of asthma in asthmatics.
Digestion problems; nausea, vomiting, diarrhea
The most severe allergic response is anaphylaxis, which may involve all of the above symptoms and can be life-threatening.

Food intolerances are food sensitivities that do not involve the individual's immune system. Many food intolerances are caused by inadequate digestion, absorption, and/or processing of the food when it is eaten. As a result, unwanted food materials may build-up, and have a harmful effect. Symptoms of food intolerance depend on the amount of the offending food eaten – the more you eat, the worse the symptoms tend to be. Symptoms of food intolerance vary and can be mistaken for those of a food allergy. Food intolerances are more likely to originate in the gastrointestinal system.
Lactose intolerance is an example of food intolerance, which occurs in people who lack an enzyme called lactase. Lactase is needed to digest lactose (a sugar in milk). Symptoms of lactose intolerance may include abdominal pain and bloating, diarrhea and flatulence.
Having a food allergy means you have to be very thorough when shopping for processed foods, as even a small amount of the food will cause a reaction. A wide range of different foods is usually processed in one plant. This means that sometimes no matter the precautions are taken to prevent cross-contamination, there is some chance that there still could have been cross-contamination. Due to this, we have to be vigilant in our label reading.
Food Intolerances means that only small amounts of the food can be tolerated if any at all without any signs of discomfort.
References:
Canadian Food Inspection Agency, Health Canada and Dietitians of Canada websites, 2007