I give this answer without any guarantees about what happens when applying from the State itself.
However, when you apply for a renewal from abroad via an embassy, they also let you keep the passport because you may well need it in the meantime (even if over a year's validity remaining).
I was told via a phone call from the Passport Office in Dublin that it was ok to travel with the old one as the details of cancellation are not passed on to any international databases as it is not stolen or lost.
The data on standard cancellations is apparently only kept on their own computer systems and the act of stamping it as cancelled was the only way anyone could know it was indeed cancelled, which would be done by the embassy when I picked up the new one.
I needed to go outside the EU. The Passport Office said this was fine on such a technically cancelled but not visibly-cancelled passport.
They also confirmed this with the local embassy. I must say I did not feel comfortable about travelling with it and thankfully the new one arrived on the afternoon before I needed to travel!