This is one of the nicer Korean restaurants I have ever been to--and it is my favorite vegetarian restaurant as well. It defines the food as "dining with a conscience." And I must say that is a pleasure to have a healthy, eco-friendly meal in a stunning environment.
Regarding the atmosphere, you must remove your shoes and sit in tradional Korean-style booths. Everything is extremely well put together, calming, and classy. I promise you will be impressed (and so will your date).
Regarding the service, I have not had better in the city. Everyone is very attentive and polite. Plus, they help add to the aura of the place through their actions. They are very careful and deliberate in the way they pace the meal, bring new items, remove items, etc. They are also happy to explain the items and offer suggestions.
Regarding the food, it is fantastic. It doesn't matter if you are a vegetarian or not. You will have a truly wonderful dining experience. I always order the multi-course meals. These consist of five or six different items, so you get a real flavor for the food. Some of the items I enjoy more than others. For instance, there is one course that reminds me of pureed baby food--but it is still good. All of the items are very flavorful, and you will walk away pleasantly full and feeling good about the food you just ate.
All in all, Hangawi is one of my favorite restaurants and I can't recommend it heartily enough.