UK and Irish citizens are allowed by the authorities to travel without passports between the two countries as part of the "Common Travel Area" between the UK and Ireland. But this does not apply to anybody else travelling between the two countries (i.e. people who are not citizens of either country).
Technically citizens do not even need a driving license for immigration, as long as they have some kind of ID to prove their identity, but in practice that's irrelevant because the airlines (as opposed to Immigration officers) insist on a driving license or passport or passport card.
I believe though that Ryanair won't take a driving license and insist on a passport or passport card. Again, that's an airline rule, not a legal/Immigration issue.