'Britain and Ireland' is ambiguous - is it meant as a geographical or a political term? Great Britain and Ireland are the two biggest islands, but this excludes Anglesey, Wight, Hebrides, Orkneys, Shetlands etc.

'The British Isles' is a geographical expression and refers to all of them. BTW they are British Isles because the Britons were there before the Anglo-Saxons.

If you don't like 'The British Isles', you'd probably be best with 'UK and Ireland'