These days, western harmony has crept into a lot of traditional music, but it wasn't always the way it is now. This seems to be a decent historical description.
http://www.tomhanway.com/melbay.htm
A few apropro quotes:
"Traditionally, piping tunes had no harmony save for droning, typically at a fifth, but even at a fourth, third, or second if appropriate to the tune, whereby tonal centers were reinforced. Fiddlers also used drones as harmony. "
"Even today, in piping, fiddling, and harping, the ornamentation and the tune are virtually inseparable. The ornamentation is the tune. "
"Some nineteenth century tunes resist harmonization or being locked into a single tonality. They have "complex tonality" with competing "tonal centers," like two (or more) little suns exercising gravitational pulls on nearby satellite notes. They give the impression of changing keys or leaving notes hanging in mid-air. Such melodies may befuddle classically trained musician whose "ears" have been trained to hear equally tempered music, major-minor keys, modulations, and Western harmony."
When you hear Martin Simpson, even on his own tunes, he's tending to try to create this kind of effect, melody, lots of ornamention (which "is the tune" according to the above), and ambiguous harmony. Sometimes the same with Pierre Bensusan, tho completely different, and he brings in a lot of jazz as well.