Carpal Tunnel Syndrome, Tendonitis and Arthritis are all very common causes of wrist, hand and finger pain. Carpal Tunnel Syndrome is a nerve problem; Tendonitis is a tendon/muscle problem and; arthritis is a joint problem. All three conditions can exist at the same time. In fact, severe tendonitis and arthritis in the wrist are recognized causes of Carpal Tunnel Syndrome.
One feature that can help distinguish between the three conditions is the presence of numbness and tingling in the fingers. Specifically, Carpal Tunnel Syndrome (which is a nerve problem) often causes numbness and tingling in the thumb, index and middle fingers, whereas tendonitis and arthritis only cause pain, without numbness or tingling.
A Hand Specialist is often able to differentiate between the three conditions based on your symptoms and physical examination.  X-rays are useful in identifying the presence, location and severity of arthritis within the wrist and fingers. MRI is helpful in identifying the presence and severity of tendonitis. Wrist ultrasound and electrodiagnositic (nerve conduction studies and EMG) tests are useful in diagnosing Carpal Tunnel Syndrome and assessing the severity of nerve compression.