Adhesive Capsulitis with Referred Pain: This condition, commonly known as frozen shoulder, is characterized by stiffness and pain in the shoulder joint. The radiating pain can be referred from the shoulder to the arm due to the irritation of the nerves or the capsule of the shoulder joint. This diagnosis is the most likely because the symptoms described are classic for adhesive capsulitis, and the radiating pain can be a part of the clinical presentation.

Cervical Radiculopathy: This condition involves the compression or irritation of the nerves exiting the spinal canal in the neck, which can cause pain and stiffness in the shoulder and arm. The symptoms can mimic those of a frozen shoulder, especially if the pain radiates down the arm.
Subacromial Bursitis or Tendinitis: Inflammation of the bursa or tendons around the shoulder joint can cause pain and stiffness, which might be confused with a frozen shoulder. Radiating pain can occur if the inflammation irritates nearby nerves.
Thoracic Outlet Syndrome: This condition involves compression, injury, or irritation of the nerves and/or blood vessels (arteries and veins) in the thoracic outlet area, which lies between your collarbone (clavicle) and first rib. It can cause pain in your shoulders and neck and numbness in your fingers.

Cervical Spine Fracture or Dislocation: Although less common, a fracture or dislocation in the cervical spine can cause severe pain and stiffness in the neck and shoulder area, potentially radiating down the arm. Missing this diagnosis could lead to serious complications, including paralysis or death.
Spinal Cord Compression: Conditions like spinal stenosis or a herniated disk can compress the spinal cord, leading to pain, numbness, and weakness in the arms and legs. This is a medical emergency that requires prompt attention.
Pancoast Tumor: A type of lung cancer located at the top of the lung, which can cause shoulder pain that radiates down the arm due to the involvement of the brachial plexus. Early diagnosis is crucial for treatment.

Brachial Plexitis (Neuralgic Amyotrophy): An inflammatory condition that affects the nerves of the brachial plexus, causing severe pain, usually in one arm, followed by weakness or paralysis. It's a rare condition but should be considered in the differential diagnosis due to its distinct presentation.
Sickle Cell Crisis: Although more commonly associated with pain in the back, chest, and extremities, a sickle cell crisis can occasionally present with pain in the shoulder and arm due to vaso-occlusive episodes affecting the blood supply to these areas.
Polymyalgia Rheumatica: An inflammatory disorder that causes widespread muscle pain and stiffness, particularly in the shoulders, neck, and hips. While it doesn't typically cause radiating pain, it could be considered in the differential diagnosis for its potential to cause significant shoulder pain and stiffness.