I've stayed at the St Regis and eaten at the Hyatt. I would chose the St Regis for an anniversary trip.

That said, depending on the time of year you go, the weather can be nicer on the south side of the island where the Hyatt is located. We stayed the first week in January and the weather at the St Regis (north Side) was overcast and cool most of the time we were there. However, the weather on the south side was usally sunny and warmer.