Diesel, while it is all inclusive, more often connotes a term for over-the-road fuel, while #2 fuel oil connotes a fuel to a specific distillation curve.
Over the road fuel basically starts as #2, but is blended and has additives that make it different from a straight run #2 distillate.
It is blended for climatic conditions. Diesel fuel in the north of the USA may be a significant percentage of #1 in order to give it a higher cloud point. In the fartherest northern regions "Diesel" might be straight #1.
There are also cetane enhancers added and pour point supressors, and other things blended in by the refinery depending upon the market (winter or summer) that it is going into.
Trucks (that I owned) that came up from the south with fuel blended for that region that got caught in a sudden temperature drop have been known to have had to be towed to an enclosed shop and thawed out and the fuel drained and replaced with fuel suitable for the ambient conditions.
It gets much more technical than that, but those are basics.
In the southern part of our country where I live, the pipelines right now are full of fuel blended for winter conditions on its way to the north for the winter and next spring's planting season. In the middle of next winter, their summer fuel will be in the pipelines. In the days before they colored 'off road' diesel to diferentiate it from 'over the road' diesel, certain marketers used to color their product to indicate that it had the winterizing additives.
For those local marketers who pulled their product from the pipelines (as opposed to those who pulled from refineries in the area) you could buy the prettiest red winterized T***** diesel in the middle of the summer here in the deep south. Ran like crap. It was too thin.
Buy from them in the winter, and you were lucky to get your vehicle started on the summer fuel that they sold.
None of these issues would matter at all with home heating oil, which is more of the straight #2 cut.
rmw