alt Hacker NewsMy understanding is that wax motors are based on actuation - they are intentionally and deliberately either heated to extend or cooled to retract, and if ambient temperature is used as one endpoint, the other will not be ambient (i.e. a heating element or cooling peltier).
Thermostatic valves are ambient at both sides - they expand and contract naturally with the existing temperature, so they are not motors per se, because there's no actuation beyond the temperature.
Why does there need to be a thermal gradient? It seems like the action driving the actuator is the expansion of the wax due to temperature. You don't need a persistent heat flux to get that, the wax just needs enough heat warm up and change phase.
Edit: also, the actuation from the wax motor only happens while it is absorbing (or losing) heat, even if it later ends up in a steady state with no heat flux.