Main advantage is that you can control it from outside the home, which is a selling point of the Nest. That's actually the main point of the app, since the device itself also has a UI that's probably easier to use if you just want to adjust the temperature.
I used that indirection to turn the heat up for my cat while on vacation, knowing that a big ice storm was coming. Couldn't do that if it only worked on the LAN.
Actually puzzled : Why would the cat need it warmer indoors if there's an ice storm? Isn't keeping it at the same temperature enough? Or is the problem because the heating switches off overnight, so you feel that it needs a boost during the day to compensate? Isn't that something that a smart thermostat should cope with?
What possible consumer benefit could that indirection have?
edit: The reason I ask is that I'm trying to figure out why an exact copy of Nest except that it didn't leak data and didn't require other people's servers to work wouldn't eat Nest's lunch.