The ENERGY STAR Connected Thermostat specification requires that the providers of the thermostats gather data about how the products are used in homes, to analyze and aggregate as a way of showing that their product saves most consumers energy. (EPA does not collect any information about single consumer's home, only averages over large groups of homes). Most of the products that can do this are WiFi thermostats, though thermostats that use other types of communications, like z-wave, can also be included.
Most homeowners can save energy with programmable and manual thermostats, if they use them carefully. However, a long history of research eventually showed that most people didn't use them that way - so EPA stopped letting them earn the ENERGY STAR in 2009. In contrast, the results for connected thermostats, using information about how customers actually use them, demonstrate that the savings will largely be realized.