Propeller engines aren’t actually “just” propellers; the important part is what actually drives the propeller, and for larger commercial planes propellers are driven by turbine engines that aren’t that different from jet turbines.
In any case, regardless of whether it’s a turbine or a reciprocating (i.e. like the one in a car) engine, they function by taking in air from the atmosphere, adding a little fuel, compressing the mixture, igniting it, and controlling how the gas expands so as to create rotational power and/or thrust.
Rockets work completely differently as they have their own sources of fuel *and* oxidizer on-board (there’s no oxygen in space, so if you want combustion you have to bring it with you). For rockets, fuel and oxidizer (together, propellant) are mixed and ignited and then expanded at extremely high velocities through a nozzle, which is then pointed in the opposite direction that you want to go in order to provide thrust. There’s no real control in the expansion beyond vectoring the thrust, whereas in other combustion engines you typically get some energy back at the expansion phase that you use to run other processes (i.e. driving the compressor fans in the turbine, keeping the engine running between combustion cycles in reciprocating engines, driving smaller generators that provide electrical and hydraulic power for both, etc.).
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