Sure you can use these differentials to generate small amounts of power but that is not always helpful.
Anything that extracts energy surely causes resistance to the flow of heat. When you're trying to suck heat out of electronics then that thermal resistance is a Bad Thing since it slows down cooling. To compensate you need a better heat sink.
That means you get a tiny amount of usable energy, but probably need bigger heat sinks. If your system has a fan then you'll need to run the fan more often too - probably using orders of magnitude more power than what you're getting.