Food allergies and food intolerances are both types of food sensitivity. When someone has a food allergy, their immune system reacts to a particular food as if it isn't safe. If someone has a severe food allergy, this can cause a life-threatening reaction. This means that people with food allergies, particularly in the case of a peanut allergy, need to be extremely careful what they eat.
Food intolerance doesn't involve the immune system and is generally not life-threatening - but if someone eats a food they are intolerant to, this could make them feel ill or affect their long-term health.