What are the differences between Irish music and Celtic music?
Of course, this might be tricky to answer, since there doesn’t seem to be a firm definition of what Celtic music is and isn’t, but I thought I’d ask anyway.
Irish music is Celtic. And Celtic music is neither. 
poodle-dog
This question is like this one:-
What is the difference between peaches and stone fruit?
The peach is a stone fruit but “stone fruit” is a category idea that covers other fruits that have some commonality but it is not an actual tangible thing.
You cannot compare a peach to stone fruit because a peach is a stone fruit. Howver, you can examine what qualifies to be a stone fruit to see if peaches, plums, nectarines and apricots fit the idea.
As denny said the poodle is a dog. What is the difference between it and It?
What is a “firm definition” of dog? From which angle will you judge firmness?
If I don’t have the academic wherewithal to firmly define dog will it raise a doubt about the dogginess of a poodle?
all poodles are dogs
not all dogs are poodles 
all dogs piddle
so do cats 
Celtic music is a puddle of piddles from various types of traditional dogs, and lately, also, from cats masquearding as dogs and also from animals of dog heritage now who prefer to meow.
do I hear a cello? 
The word “Celtic” has various degrees of precision and vagueness, depending on who is using it.
If you are talking about prehistoric culture, I have seen it argued that all of Europe is Celtic, beginning with the Greeks, as Keltoi. The Aryan invasion of India brought Celtic culture to India. There are indicators of this in the veneration for cows, the stratification of society, and linguistic considerations. So the “Celts” would cover Europe and parts of Asia minor.
If you are talking linguistically, then you are talking about places where the languages spoken (or, originally spoken) belong to the same group. That would be Ireland, Scotland, Wales, Isle of Man, Cornwall, Brittany, parts of Spain, and Finland. You could include Nova Scotia in that, as Gaelic is still spoken there.
If you are talking about Music, the term takes on a commercial taint, as it tends to be used as a label by distributors who are less concerned about the accuracy of the term and more concerned with the sales - the number of people who will be pulled in by the term.
Considering that similar sorts of “diddley” is played by Americans, Swedes, French, Dutch, Japanese and even (gasp!) the English, it might be a mistake to confine the term “Celtic” to Irish, Scots and Brittany music.
You can tell from the attitude of the previous responses that it’s a vexed question and one which experienced players prefer to avoid. But in general, “Celtic” music spreads a wider net than Irish music. And a much vaguer net.
So tell me, what is “World Music” - and what’s the alternative (“Out of this World Music”)? A lot of the Folk I listen to seems to come under this category, even if it’s Irish.
IMHO ‘World Music’ is a catch all phrase for in general but not wholly roots music from the various continents. Irish Music is a Celtic Music and Celtic Music is a World Music.
Ah. What’s “roots music” then?
Ooh! I know! I know! It’s music played on a tuber.
Beets me.
not bad … 
What has “Asia minor” got to do with India ?
Cats do not come from dogs, at the root of it they may have shared a common ancestor with dogs who were neither dogs nor cats. It’s always been obvious that India was Celtic, what with the plaids and all.
Buggered if I know.
Just trawling though my library, the categories are
Folk
General Folk
Contemporary Folk
Traditional Folk
World Music
General World (Music, I expect)
Celtic
General Celtic
General Rock
Blues
Quirky (I may have made that one up)
Bluegrass
General Pop
General Rock
Classical
General Classical
Kathryn Tickell, who is Northumbrian, is under “General Celtic”.
Planxty’s “Cold Blow and the Rainy Night” is under “General Folk”.
Michael Ormiston’s Mongolian Throat music is under “General World”.
My own stuff (Elephant) is under “Contemporary Folk”, which is reasonable.
Oh, and there’s a category “Art Archive” under which Christian Boltanski recorded my heartbeat!
John Martyn is under “Traditional Folk” - that would not have been my decision. 
“The Bonzos” are under “Quirky”.