While the common cold is an infection from any of the many cold viruses, a nose (nasal) allergy (also referred to as allergic rhinitis, hay fever) is caused by substances entering the nose that you are allergic to. Common substances causing nasal allergies are dust, animal dander, or pollen. Symptoms can also occur when you eat a food that you are allergic to.
Similar as in the case of the common cold the symptoms are not caused by the allergens, but by the body's defense response to the foreign invaders. Perhaps not surprising than that the symptoms overlap somewhat with those of the common cold, but also include watery eyes, headache, itchy nose, -mouth eyes or -other areas. Fever is normally absent except when secondary infections occur due to the allergy.
Treatments (antihistamines, decongestants) also overlap but in the case of a nasal allergy extend further to a stronger attempt of suppressing the allergic response e.g. Nasal corticosteroid sprays.
Sinusitis is an inflammation of the sinus cavities in the face. These are normally filled with air but may become blocked with fluid and lead to sinusitis. The common cold, influenza or nasal allergies can be the cause of sinusitis, but there may be other causes like nasal polyps or a deviated septum. Bacteria, viruses, and fungi can grow on the mucous in these cavities and cause a sinus infection. Sinusitis is usually accompanied by pain in the sinuses that often increases when the sufferer bends forward or lies down.
References:
Medlineplus
Webmd
Wikipedia
