Demon Possession of Jesus' Time - Today's Diagnosable Mental Disorders? Quite a few of the things Jesus and his disciples had to address they had diagnosed as "demons". Today we call things of similar symptoms "disorders".

Epilepsy: Matthew 17:14-16

Multiple Personality Disorder: Mark 5:1-20

For lack of time I cannot give an exhaustive list, yet its easy to find quite a few. Yet I see that "pharmaceuticals" and other drugs tend to be the fallback and treatment. 

What connection, if any, exists between the ancient diagnosis and modern ones? Is there any distinctive difference between them?
I'm going to self identify for the purposes of this answer. I have a multitude of mental illnesses. Schizoaffective disorder (hallucinations of all types, delusions of several types, mania and major depression) and OCD as well the lesser subsyndromal OCD (obsessive behaviour and intrusive thoughts). On bad days the Dyslexia (trouble with language) and Autism (experiencing reality differently) are also a problem. All my symptoms are under control though sometimes I still experience hypomania and depression but not mania and major depression.
I understand your concern that I might be demon possessed however such concerns are not realistic. My brain is broken the methods by which it is corrected is a medication by no biblical definition would it fit demonic possession.

Luk 13:16  And ought not this woman, being a daughter of Abraham, whom Satan hath bound, lo, these eighteen years, be loosed from this bond on the sabbath day?

Mental diseases are no different than physical diseases. They are satanic bondage and oppression/bondage/affliction is not possession. The woman in Luke 13 was bound by the devil and a child of God. This is why:

pos·ses·sion pəˈzeSHən/Submit noun
1.the state of having, owning, or controlling something.
2.an item of property; something belonging to one.

-Google dictionary

Mar 3:25  And if a house be divided against itself, that house cannot stand.

Demon possession is the opposite of salvation. It is to be owned and controlled by the devil. To be filled with the devil in the same manner in which those who are saved are filled with the Holy Spirit. Spiritual houses are not divided against themselves either you belong to God or you do not.

2Co 10:5  Casting down imaginations, and every high thing that exalteth itself against the knowledge of God, and bringing into captivity every thought to the obedience of Christ;

Those who have religious intrusive thoughts (unwanted random thoughts generated by broken neural pathways about religious figures) and personality disorders have to put up with things which would scare normal Christians. Just because our burden is greater does not mean we belong to the devil. It means we must have medication to help us cast down that which does not conform to the word of God.

Job 2:9  Then said his wife unto him, Dost thou still retain thine integrity? curse God, and die.

Your tempted to commit adultery; we're tempted to commit adultery. We're tempted to curse God and die; you're not. Our temptations and faults don't make us evil in the manner of demonic possession.

Jas 5:11  Behold, we count them happy which endure. Ye have heard of the patience of Job, and have seen the end of the Lord; that the Lord is very pitiful, and of tender mercy.

We just are more patient than you.