I've never been to Kings Bay. Winter has got to be better.
Last I was in Groton, getting an appt at the hospital was damn near impossible. If you've got kids that might be important for you to know.
Hubby says Kings Bay is newer, more amentities on base. Seems that it's in the middle of nowhere, so anything off base is a drive. Groton has shopping centers near the base, within just a minute or two. A mall is further away, across the bridge into New London. Maybe 15-20 minutes away.
Hope that info is helpful.