Between tegretol and trileptal, what is the best treatment for chronic neck and hand pain?

Neither. Neither would be high on my list. What is the cause of the pain?
Depends. There are many causes of neck and hand pain. Sometimes a pinched nerve in the neck causes the hand pain. You may need an MRI and possibly electrodiagnostic studies of the arm to get more information. There are many other treatment options.
If... ...Both have been shown to be effective for a condition, trileptal is less likely to have side effects--but this does from person to person.