"turboprop" and "diesel" are really contradictions, as they generally refer to totally separate types of engines.
As for similarities in the jet engine (or gas turbine, or Brayton Cycle engine) with an engine running on the Diesel Cycle, I'm not sure you could classify a continuous combustion (turbine) engine as being the same as a compression ignition (diesel) engine in terms of igniting the fuel, but the thermodynamic cycles do have some similarities.
As for the fuel question, as L-188 said, both diesel fuel and jet fuel are basically kerosenes. Both have flashpoints that are much higher than gasoline (IIRC, gasoline has a flashpoint of around -40C (-40F), which is why it's so dangerous, whereas kerosenes are around 40C (~100F)) (also note that flashpoint is the lowest temp that the vapours can combust, but is not the same as autoignition temperature).
Diesel engines tend to be heavier than similar spark-ignition (gasoline) engines, and usually operate at lower speeds, but are longer lasting, have better torque curves, and are able to burn different fuels. The last reason is the biggest part of why general aviation aircraft are beginning to look at diesel power.
Further to the similarities of fuels... saying "jet" or "diesel" fuel has no real meaning beyond application - the exact composition will vary depending on the conditions they are blended for use in. For instance, I've heard of turbines burning diesel, and if you look up acceptable diesel fuels in a heavy equipment operators manual, it will list a variety of fuels that can be burned (IIRC from a certain Caterpillar diesel engine, I seem to recall a couple of grades of "diesel fuel," as well as a few kerosene derivatives). For that matter, diesel fuel sold here in western Canada will vary in composition by season to prevent it from "gelling" (basically freezing) in the fuel tank.