I am not sure if we are answering the right question here--is it the actual music that we are trying to distinguish, such as an Irish Band Playing, as opposed to a Klezmer Band playing, or are we trying to figure out how to tell if a tune, which I take as a melody, is an Irish one or a Klezmer one?

The first question is a relatively easy one to answer-- since Klezmer music tends to be polyrhythmic and Irish Music tends not to be. This simply means that there is an strong underlying beat, but that there is also at least one other beat that plays "against" it.(When one voice is playing, the other beat is implied). Also, the element of improvisation(as is pointed out above by Vincenzo), keeping in mind that the improvised voice works off of a counter rhythm, as you might hear in Jazz.

It is no coincidence that many Jazz greats, paricularly composers, came from Jewish musical traditions--Klezmer clarinet and Jaxx clarinet styles are fundamentally related--("And the Angels Sing" is actually a Klezmer tune, and one very famous recording of it jumps from Jazz to Klezmer and back).

The basic Klezmer beat (dumDa-dum-dum-daadum) has always seemed to me to be essentially a quick version of the Arabic Rhumba, though the melodies and improvisation often seem to be directly based on turkish classical music.

The Klezmer band seems to be based on a version of the Fazil or turkish classical ensemble (which is, functionally, much more like a western jaxx ensemble than like a western classical ensemble), and the traditional Klezmorim often included a Tsimbaly, which was a Santour, or hammered dulcimer.

However, when we get down to tunes, the distinctions are harder to make. There are certain melodies that seem to pop up in all kinds of music, and if you listen much to either Irish or Klezmer music, you will know that the musicians have no hesitation about taking melodies from other genres and adapting them--the results can be beautiful or hysterical--