In terms of internal conditions, diesel engines operate in an entirely different performance envelope than most gas engines. The extra abuse they place on components, combined with the chemical makeup of the fuel itself, makes using the right oil crucial for engine performance and longevity.
Diesels don't ignite their air-fuel mixture with a spark, they just squeeze the mixture until it explodes with heat and pressure. This not only places a great deal of stress on internal components, it tends to atomize impurities (like sulfur) in the diesel fuel and transfer them to the oil.
Many diesel-specific engine oils contain lubricants like ZDDP (Zinc dialkyldithiophosphate), to help keep things slick to help cope with the engine's additional stresses, and detergents that keep contaminants like soot suspended in the oil. Left to its own devices, these contaminants would embed inside the engine and build up to restrict oil flow and reduce lubricity.
Any lubrication is better than none, so you can use regular car oil in a diesel for short periods of time if you're left with no other option. The only other time you might consider running regular oil in a diesel is if you're flushing the engine, in which case you'll wind up draining it after 500 miles anyway.
