I'm a local Realtor in Newtown and used to work for an agency such as the one who is purchasing next door to you when I lived in New York. My experience is that many neighbors are not happy when a group home moves into the neighborhood and you are correct that this type of housing is exempted by the State (like a church or school).
The agency who will own the home next door to you is regulated by laws of all sorts. They must keep the home up and may even make upgrades to the home that will increase its value. Most of the time, these homes do not stick out like a sore thumb.
You can petition the state for a tax break; however, it is not the state who determines the taxes on homes. It is the local government.
Prudential CT Realty - Newtown