Day nursery usually provides childcare for longer hours, and may take children younger than 3. They also tend to run through school holidays and their primary purpose is childcare for (working) parents. Whilst they will follow the foundation stage curriculum they may only do so for part of the day.
Pre-school isn't a formal term around here, and can be anything from a playgroup operating for 2.5 hours per morning to a more formal nursery class attached to a school. The difference seems to be here that pre-schools stick to school holidays and aren't as much use for working parents.
Ds2 goes to 2 "pre-schools", one of which runs in a church hall for 2.75 hours per morning, the other is attached to our primary school and is open 8am - 3pm during term times.