That's correct, but people from other countries mix them up all the time. Over here, Irish is just called Irish or Gaeilge, never Gaelic or Irish Gaelic. The funny thing is, Gaelic over here means Gaelic football.
But I heard some Scottish Gaelic speakers don't like their language to be called "Gaelic" either. They like it to be called Gàidhlig or Gallic.
"The Irish spirit can't be enshackled in the English language. That's why we swear so much".