I would agree with Suzy that Hampton Bays and Quogue are completely different towns.
Quogue is a very quiet old Hampton's Estate living. It has a very quaint village Main Street, some beautiful private village beaches and a great small elementary school. The housing market trends on the higher price side for the "western" hamptons because of the low village taxes and exclusive neighborhoods.
Hampton Bays is the working town. There are a lot more year around residents with the largest shopping hamlet for the area. Taxes are slightly higher then Quogue because they are not in a Village but housing prices are lower. There are some very beautiful parts of Hampton Bays with some great waterfront locations.
East Quogue is the best of both towns in my opinion. It has a great small town USA Main Street with some wonderful shops. The elementary school is one of the best in the area and is part of the Westhampton Beach High School district. Taxes are slightly higher then Quogue but lower then Hampton Bays. Housing prices tend to be higher in East Quogue then Hampton Bays but less than Quogue for the same style house.
I would recommend contacting a local realtor like myself that lives in the area to show you around.