What are the operational differences between normal diesel fuel used in cars and the Jet-A diesel type fuel used for some prop aircraft such as the Diamond DA62? I know that aircraft use Jet-A fuel, but could an airplane be run on car diesel?
In at least some cases, yes. Continental's CD-155 diesel engine will run on auto diesel, Jet A or any mixture of both:
The CD-155 is certified for the use of both jet fuel and diesel (DIN EN590) and is running with the two fuels in any mixture ratio.
EN590 is the European standard for auto diesel. The Gemini 125 engine can run on Jet A and diesel according to their website, as can the Austro AE300. So it seems that in general, aircraft diesel engines can indeed run on auto diesel.
Yes, and diesel cars can run on aviation jet fuel. Have heard delightful testimony of this from some owners of diesel cars who worked at an airport.
The Austro diesel engine in a Diamond aircraft is an adaptation of a Mercedes car diesel engine. Runs fine on both car diesel and jet A-1.
Some more examples of (ultra-light) planes running on car diesel: