Are there any differences between Protestant and Catholic theology concerning the distinction between demonic possession from mental illness? Is there a difference between Catholic and Protestant Theology in regards to what constitutes Demonic possession as opposed to mental illness?
From this article by Got Questions Ministries, who describe themselves as "Christian, Protestant, conservative, evangelical, fundamental, and non-denominational", the answer to "How do we distinguish a psychological disorder from demon possession?" is


  The short answer to this question is that the Bible does not speak to
  distinguishing between demon possession and a psychological disorder.
  Because God chose not to equip Christians for this task, we should
  probably assume this is not something we are called to do.


In the Gospels, for example in Matthew 9, there is a clear distinction between 'every disease and every infirmity' that Jesus healed and the casting out of demons.

In the Catholic Church, "before an exorcism is performed, it is important to ascertain that one is dealing with the presence of the Evil One, and not an illness."

cf. Catechism of the Catholic Church 1673


  1673 When the Church asks publicly and authoritatively in the name of Jesus Christ that a person or object be protected against the power
  of the Evil One and withdrawn from his dominion, it is called
  exorcism. Jesus performed exorcisms and from him the Church has
  received the power and office of exorcizing.1 In a simple
  form, exorcism is performed at the celebration of Baptism. The solemn
  exorcism, called "a major exorcism," can be performed only by a priest
  and with the permission of the bishop. The priest must proceed with
  prudence, strictly observing the rules established by the Church.
  Exorcism is directed at the expulsion of demons or to the liberation
  from demonic possession through the spiritual authority which Jesus
  entrusted to his Church. Illness, especially psychological illness, is
  a very different matter; treating this is the concern of medical
  science. Therefore, before an exorcism is performed, it is important
  to ascertain that one is dealing with the presence of the Evil One,
  and not an illness.2
  
  1. Cf. Mk 1:25-26; 3:15; 6:7, 13; 16:17.
  2. Cf. CIC, can. 1172.