None, really. Gaeilge is the term the Irish language has for itself, which in turn gives us the English-language word Gaelic. Both Gaelic and Irish were used to describe the language for centuries; for example the first movement to revive the language was called the Gaelic League. Revivalists who associated the language with nationalism preferred to call the language Irish in order to give it a national image. This term is now the more commonly-used one. In English the term Gaelic applies to both Scottish and Irish Gaelic, and this can very often lead to confusion when the context is not clear. For example, the Teach Yourself Gaelic course refers to Scottish Gaelic rather than Irish Gaelic. To avoid this confusion Irish is often used when speaking about Irish Gaelic.