Babbo or Blue Hill for a Birthday dinner?
Went to Blue Hill (in town) for a birthday dinner once and found it to be just ok. I think the food is kind of dull as is the vibe. Also, the tables are very close, so close that I knocked into the table next to us when getting to my booth seat and tipped over a whole bottle of red wine into the lap of the gentleman at the adjacent table. Mortifying.
Babbo is much more bustling, loud. The food of course is terrific.
Very different places.
I love both, and side with the first posters' logic of apples and oranges. These two popular restaurants offer two different experiences.
Blue Hill prides itself under cooking with organic ingredients (many of which from their farm Stone Barns upstate). Many meats, vegetables, and fruits come directly from the barn and you can taste the difference. It offers excellent New American, some of the best in the city. If I were to say what's better here than Babbo, it would be the desserts, by far. I've eaten at both places about 6 or 7 times each, and am consistently impressed by their pastry chef; chocolate bread pudding and blackberry cheesecake are particularly memorable. Their menu is always seasonal and always changing, to maintain freshness though they do have plenty of staples; it's fantastic. The vibe of the room is sophisticated, small but paradoxically plenty of room. Waiters are intelligent and not fussy.
Babbo, however, is arguably the best Italian restaurant in the United States, with a larger menu than Blue Hill's. It's louder, more boisterious than Blue Hill, and they are always playing some fun music. I've always felt that Batali's biggest strength are his pastas, which successfully fuse traditional and contemporary italian cuisine so effortlessly. You can't go wrong with their pastas, their bolognese is authentic, their beef cheek ravioli with oxtail ragu and black truffles is a modern-day classic. If you want a real trip, go with the pasta tasting menu. Their regular tasting menu isn't as mind blowing to me, still pretty good though. Service is solid.
So there is my brief take. hope it helps. Both are fantastic though, they are both some of my two favorite restaurants in the city. Very different, but both will give you a fantastic meal.