Celtic is the broad family — languages and cultures including Welsh, Breton, Cornish, Irish, Scottish Gaelic and Manx. Gaelic is a branch inside it, covering Irish, Scottish Gaelic and Manx specifically. So all Gaelic languages are Celtic, but Welsh is Celtic and not Gaelic.
The core difference
One is a family, the other a branch within it. The relationship is the same as between "Romance" and "Italian".
- Celtic — the whole family, split into two groups: Goidelic (Irish, Scottish Gaelic, Manx) and Brythonic (Welsh, Cornish, Breton).
- Gaelic — the Goidelic branch. Irish, Scottish Gaelic and Manx.
Where the confusion bites
Calling Welsh "Gaelic" is a straightforward error — Welsh is Brythonic. And "Gaelic" alone is ambiguous between Irish and Scottish Gaelic, which is why careful writing names which: Irish (or Irish Gaelic) and Scottish Gaelic. In Ireland the language is usually just called Irish.
Pronunciation
Scottish Gaelic is generally said "GAL-ik"; Irish Gaelic is generally "GAY-lik". The football club Celtic is "SEL-tik", while the language family is usually "KEL-tik" — a split that catches people out constantly.
Frequently asked questions
Is Welsh a Gaelic language?
No. Welsh is Celtic but belongs to the Brythonic branch. Gaelic covers Irish, Scottish Gaelic and Manx.
Does Gaelic mean Irish or Scottish?
It can mean either, which is why careful writing specifies. In Ireland the language is usually called Irish; in Scotland it is Scottish Gaelic.
Is Celtic pronounced with a K or an S?
The language family and the peoples are normally "KEL-tik". The football club and the basketball team are "SEL-tik".