The sulphur did several things, but one of the primary ones was that it lubricated the injection system.
The epa came out and demanded much less sulphur, to lower emissions, and vehicles started needing injector system parts sooner than before, as they would wear quicker.
Newer models should be designed to run the lower sulphur content fuel, so this should no longer be a problem.
Diesel fuel is hygroscopic, and will absorb moisture straight from the atmosphere and hold it. Water content, as it gets higher, will cause many problems, including rusting in tanks, and it wont burn as well. There are additives to reduce this, and many, if not most vehicles have a water seperator in the fuel system, either stand alone, or a combined fuel filter water filter.
Also, when it gets real cold, the wax still present in diesel can harden and separate out, called gelling. This thickens the fuel, and it wont burn. It can be reclaimed, but takes time, and a larger tank. In a vehicle, its usually just flushed out.