The Nest thermostat saves you money because of it's "intelligence" It learns how long it needs to get from it's present temperature to the required temperature, it monitors this continually. The need for wifi is so it can get weather data for your postcode/zip code which helps with it's decision making.
Due to this knowledge it will be able to either come on early or later due to weather conditions.
As it knows how long to change temperature it can also switch off early because it knows that there is enough heat already in the system to raise the temperature up to the desired setting.
A normal room stat set at 22c or 70f will switch off at 22/70 but temperature can rise another 1,2 or 3 degrees c/8f, that's a fair amount of energy saved.
You can set a minimum temperature for away mode and you can set another minimum temperature for when it's off i.e. 18c in away and 9c when off (frost protection etc)
Nest will learn your routine to some extent by the changes you make to the settings.
I installed one early November and we can see the difference now in comfort and costs. We have turned it down from 24c to 22c at peak times and it settles at 20.5 for most of the evening when my wife is watching TV
Just one other thing, you don't need wifi to use it, wifi is there for remote control and weather data and the wireless connection is just to control the system by the thermostat. You can override if you want by pressing the front of the heat link if you're having problems.