The first major difference between diesel engine oil and conventional oil is the presence of additives. To clarify, both kinds of engine oil are packed full with various chemical additives to reduce emissions and improve reliability. This includes stuff like corrosion inhibitors to prevent the creation and build-up of rust, anti-wear agents to keep the engine from damaging itself as it runs, and detergents to remove gross soot buildup, to give a few examples.
The big difference is the quantity of additives present in diesel engine oil versus the regular stuff. Since diesel engines are larger on average and create more sooty gunk in general operation, diesel engine oil has a greater concentration of the relevant additives to make up the difference. If you tried to use oil with the same level of additives as diesel oil in a regular combustion engine, it would be completely flooded with overwhelming levels of detergents, which could compromise the engine’s structural integrity and cause it to belch smoke.