Advantage: AvGas is heavily taxed in Europe. This is primarily an envy tax, because it brings in less than the cost of its administration, but the non-flying majority feels good by "punishing the millionaires who waste their money on flying". AvGas costs currently about 2.6 - 3 € per liter in Germany, for example.
Jet fuel cannot be taxed in the same way because arbitrage is too easy. Airlines would simply stop refueling within Europe. At prices of around 1 € at large airports the difference is substantial. With a DA-40 you might not get the same price that airlines get, but a sizable difference remains.
Disadvantage: At pure GA airports, you will have a hard time to get Jet fuel at all. There are very few piston engines which can use Jet A-1 and are rated for aviation. Car engines cannot be used without heavy modifications, because they are not designed for continuous operation at 60% or 70% of their maximum performance. Due to the higher internal pressure, the engines tend to be heavier and produce more vibrations. However, since the specific fuel consumption of a Diesel engine is lower, the total system mass for ranges in excess of 1000 km should be lower.
The GA market is simply too small to support the design of new engines. We have to use what was created half a century ago. It's a shame, but it has been shown several times in the last two decades - there were a number of attempts to create GA diesel engines, with very little to show.
In the 1930's Junkers made a range of very interesting Diesel piston engines (see here) which powered a range of aircraft designed for very long ranges. They were used for transatlantic mail service (see here and here for examples).