LRR: Thanks for that! I had no idea that "on the weekend" would be wrong in British English!
YL: During the weekend makes it sound like to goes on for a long period. Going out to eat may take a long time, but not take the entire weekend. It's not a requirement that something last the entire weekend to use "during," but it does carry an implication of a longer activity or status than dinner.
N2G: Yes, that's fine. There's no real difference.