Where is Byzantine Music used?
Generally, Byzantine notation is used in churches near (or by people who are from churches near) Constantinople, generally the Greek Orthodox and the Arabic Orthodox. The Serbian Orthodox use a slightly different system of music; the Russian Orthodox use what is essentially Western notation and music. Also, it is worth mentioning that some Eastern-Rite Catholics use Byzantine notation.
What is Byzantine Notation - a comparison
Byzantine notation is very different to Western notation in most ways. A comparison will be done for the benefit of most readers (who, it is assumed, do not know Byzantine notation, but are familiar with Western notation).
Western notation is based on a staff, where the pitch is determined from where the note is on the staff (regardless of the previous note). Byzantine notation, on the other hand, is relational; the note is dependant on the previous note and the symbol itself, which specifies how far the note is from the previous.
While there are differences in speed and in whether a certain note should be flat or sharp in Western music, Byzantine music has this down to a (very complicated) artform, using certain tones which always have a specific note being sharp or flat.
One near-similarity is the scale. In Western music, 'Do' (the start of the western scale) corresponds to the Byzantine note 'Ni', which is a note below the start of the Byzantine scale.
Byzantine music has eight tones, with which aims to evoke certain emotions (for example, tone 5 is designed to be a victorious, almost arrogant, tone; while tone 6 is designed to be a mournful tone). Also, much of Byzantine chanting can be done without use of music, due to the use of original melodies (gk: αυτόμελον). While there are tens of thousands of hymns in Byzantine music, they are all based on less than two hundred original melodies.