Trigeminal Neuralgia: This condition is characterized by sudden, severe, and stabbing pain typically around the eyes, lips, or nose, which can be triggered by light touch. The pain is usually unilateral and can be described as throbbing or electric shock-like. Given the description of "left throbbing severe facial pain," trigeminal neuralgia is a strong candidate due to its classic presentation of severe, unilateral facial pain.

Migraine: Migraines can cause severe, throbbing pain, often unilateral, and can involve the face. They are frequently accompanied by other symptoms such as nausea, vomiting, and sensitivity to light and sound.
Cluster Headache: These are extremely painful headaches that occur in cycles or clusters, often waking the patient from sleep. The pain is typically unilateral and centered around the eye, with associated symptoms like nasal congestion and eye watering.
Temporomandibular Joint (TMJ) Disorder: TMJ disorders can cause pain in the jaw, face, and surrounding areas, which can be severe and throbbing. The pain can be exacerbated by jaw movements.

Acute Angle-Closure Glaucoma: This is an ophthalmic emergency that presents with severe, unilateral eye pain, often accompanied by blurred vision, nausea, and vomiting. It requires immediate medical attention to prevent permanent vision loss.
Cavernous Sinus Thrombosis: A rare but potentially life-threatening condition that can cause severe facial pain, especially if it involves the trigeminal nerve distribution. It often presents with additional symptoms like fever, ptosis, and ophthalmoplegia.
Giant Cell Arteritis: This condition can cause severe, unilateral headaches and facial pain, particularly in older adults. It's crucial to diagnose promptly to prevent complications like vision loss.

Raeder's Syndrome: A rare condition characterized by a combination of facial pain, ptosis, and miosis, typically without anhidrosis. It's often associated with a lesion in the middle cranial fossa.
SUNCT (Short-Lasting Unilateral Neuralgiform Headache Attacks with Conjunctival Injection and Tearing) Syndrome: An extremely rare condition that presents with very short-lasting but extremely painful headaches, typically around the eye, with associated conjunctival injection and tearing.
Paratrigeminal Syndrome (Raeder's Syndrome): Presents with a triad of miosis, ptosis, and facial pain, usually due to a lesion affecting the sympathetic nerves to the eye and the trigeminal nerve.