Here is an answer describing this topic a bit: Is it possible to build a thermoelectric nuclear power plant?
The phenomenon that makes it possible is the thermoelectric effect. And the object is called a thermoelectric element or module.
Basicly, heat up one end of a metal bar and electrons obtain more kinetic energy there and vibrate more and move faster around - and "fill" more. They will start drifting towards the end where they fill less. And drifting electrons is current.
The "push" that makes them want to move is of course caused by the temperature difference, yes. And that "push" is called the Seebech voltage $V_s$:
$$V_s=S\Delta T$$
$S$ is the Seebech coefficient (depends on charge carriers and their mobility and other detail of the material.)
The big issues are two:
We of course have to set up a circuit to make current run continuously. But here we get a charge-carrier problem, since electrons repel the hot side and won't flow back towards it in the circuit. Therefore other charge-carriers are necessary in half of this circuit, and semiconductors are usually used for that.
The driving "force" is the temperature difference, yes. Not the temperature itself. The hot side is not more important than the cold side and the actual values are irrelevant. A huge issue is the simple fact that natural effects like heat conduction, Joule heating (when stuff heats up because a current runs through it) and others are heating up the cold side and maybe cooling down the hot side.
Issue 1 requires more complex materials and issue 2 requires more complex material structures (you can fiddle with the crystallic lattice structure to reduce the thermal conduction coefficient etc.) All in all, suddenly this simple phenomenon turns out hard to apply. Making a good, commercially viable device requires quite some work and research.
Sitting with that power pot out in nature might provide a cold source as convection from wind, and such natural sources of hotness and coldness are very helpful to make it work properly and overcome the heating-issue.
Putting a thermoelectric element on car exhaust pipes and alike are applications undergoing huge research these days. Though as far as I know the technology/phenomenon is not given much attention or research compared to other fields, unfortunately, mainly because it is still young.