For clarification of the answers above:
The UK and Ireland have a 'open border' policy, allowing free movement of citizens of the two countries across the border. (The concession does not apply to British citizens born abroad, nor to citizens of other countries). So, as far as 'the authorities' are concerned, Mrs D is correct (assuming, of course, that you are both British-born UK citizens).
However all airlines require passengers to provide some form of photographic ID. Ryanair ONLY accepts passports as ID for ALL of its flights (even those which are entirely within the UK) but all other airlines flying between the UK and Ireland accept other forms of photo-ID, such as driving licences and bus passes. e.g Aer Lingus:
http://www.aerlingus.com/travelinformation/knowbeforeyoufly/visaspassports/#d.en.4568
Ferry companies operating between the UK and Ireland also accept a wide range of photo-ID. e.g. Irish Ferries:
http://www.irishferries.com/uk-en/faq/passports-identification/
Similarly, British citizens entering Ireland by rail (from Belfast) are advised to carry photo-ID but passports aren't required.