I will have a go at this although I am fairly new to the Greek Language and I am sure our teacher will say if I am incorrect!
Καθαρεύουσα was a 19th century mixture of ancient and demotic Greek and I think it had all sorts of 'breathing points' unlike the current demotic language which has a 'tonos'. In the 20th century it was used mainly in government and legal and I think (and I remember this because I was attached to the Greek Air Force for a few weeks in 1969 with the Royal Air Force) that during the time of the Colonel's dictatorship from 1967 to 1974 it was brought back as the compulsory language for a while?
On the return of a democratic government in Greece I think everything reverted to 'demotiki' and one of the Greek Prime Ministers in the 80s declared it official.
If I am incorrect in any of this I apologise