I love playing a fiddle tune that might be from the Southern Appalachians then following that up with a jig or reel that may have originated in West County Clare, for example. I do so freely with little concern for authenticity. Heck, I play tenor banjo so what does it matter?!
The way I see it, tunes are just a combination of notes and all I'm trying to do is make it sound like the way it’s supposed to and/or the way I want it to. Nonetheless, it doesn’t hurt to be aware of some of the melodic characteristics found in tunes of Appalachian origin vs. those of Irish origin. These include:
Notes: Irish tunes can be rather notey. Anyone who has tried to play an Irish reel at a blazing speed knows this! In an old-time tune feel free to leave out more notes in favor of (deceptive) simplicity.
Ornaments: Use triplets to get that Irish sound. For more of an old-timey sound try using double stops.
Melody/Rhythm: On old-time tunes you may want to bring out that boom-chucka/shuffle rhythm. For Irish, focus on the melody, but make sure you can distinguish between a jig, slide and slip-jig, and a reel, hornpipe and polka. Each of these tune types has its own rhythmic emphasis.
Clarity vs. Dissonance: On Appalachian tunes you can use more dissonant notes and quarter tones. For Celtic tunes emphasize fluidity and note clarity.
My two cents: It’s a non-intellectual process. If you use your ear to pick up on the nuances of each individual tune you'll already be making the distinctions you're supposed to, regardless of tune type or country of origin. Just let it happen.