Both Smith & Wollensky and Del Posto are very expensive and if you decided you wanted to spend that much money, I'd pick other more romantic restaurants:
One if by Land, Two if by Sea Youd need a cab downtown, but Aaron Burrs 1767 carriage house is always at the top of the lists for NYC romantic restaurants. At one point, more marriage proposals had taken place here than any other restaurant in NYC, have no idea if thats still true.
http://www.oneifbyland.com/
The Four Seasons Restaurant - Ask to dine in the beautiful pool room, it is very close to your hotel and is NOT part of the Four Seasons Hotel several blocks away.
http://www.fourseasonsrestaurant.com/index2.htm
My personal favorite romantic restaurant in the city and worth every penny
La Grenouille. It is truly one of the most beautiful dining rooms around and they treat everyone, even first timers like royalty.
http://www.la-grenouille.com/
If those are above your budget, look at these:
Erminia - cozy Italian considered one of the ultimate date restaurants
http://www.erminiaristorante.com/
Jo Jo - A Jean Georges restaurant in a beautiful townhouse thats much less expensive than going to THE Jean Georges restaurant.
http://www.jean-georges.com/
And, if you are even more budget minded, then I'd go to
Vivolo. A romantic Italian in a townhouse with a fireplace on both floors. The ground floor is wood paneled, cozy and sort of clubby while the upper floor is more like a formal parlor. Both are nice and food is good. Tell them you are celebrating an anniversary and request a table near one of the fireplaces.
http://www.vivolonyc.com/
Hope these help, there are SO many restaurants in NYC, it's hard to try to give recommendations because there are so many great choices!
