The two elements of "smart" are often combined, but there's no need to have both.
There's remote control, which is one thing. My Honeywell system can do that, but even if I don't connect it to the WiFi (and earlier versions required extra hardware) it'll still do smart local control. I lose weather compensation in that case, though, as it uses a remote forecast to predict future demand and not bother turning the heating on if it's not actually going to be needed.
The more important aspect of "smart" heating is proportional control and predictive heating. As you say, regular thermostats will turn the heating on when the temperature is below the set point, then off when it's above, with some hysteresis to avoid flapping. And one function of the smart system is to predict how early it needs to turn the heating on to get up to temperature in time. The Honeywell will also switch the heating off early if it's confident the temperature will drop by less than 0.5°C before the end of the set time.
Beyond that, though, the controller will modulate the boiler to maintain the set point -- heating systems term this "TPI", I assume it's not a PID controller because it lacks the derivative component. Time Proportional and Integral either (in my old house) controls the boiler modulation directly or (in my new house) turns it on for a few minutes in every time slice in an attempt to get the right amount of heat into the system to maintain the set point.
Where a system like the Honeywell really shines though is that I'm not turning the whole house on or off -- I've smart TRVs on each radiator, so I can have heat in my office during the day and in the lounge in the evening and not have to keep rooms warm when I'm not using them. Having recently moved house and consequently had a few weeks without the smart system I can attest that it really does make a big difference.