According to Collins dictionary they both derive from the Latin districtus, meaning divided in mind
The OED gives as one definition for distress:
2. a. The sore pressure or strain of adversity, trouble, sickness, pain, or sorrow; anguish or affliction affecting the body, spirit, or community
which strikes me as apt for the disease of schizophrenia. Could a new name be developed from the etymology?
Districtus does not sound right, because of the implication of physical spasm or rictus. But how about distrenia, or dystrenia? It is also reminiscent of the mysteries of dystonia
It is not in the OED & Google found no mentions.
Link: Irish jokes & schizophrenia