Same ADN, but different perfumers and different fragrances overall.
DHI is not only sweeter, but sexier. Too playful for a formal environment I'd say, perfect for a night out, a date,... if it's a not very hot summer night I think you can pull it off, just be careful with the trigger. I'm talking about the pre-reformulation formula, but the new one I've heard is very close though.
DH is more formal, slightly more elegant, more versatile. I love the newest formula, much more wearable.
Own and love both.