Explosive rounds increase the damage somewhat. As enemy planes are usually armored it's worth loading one of the guns with some armor piercing rounds. Dum dum bullets pack a wallop but don't really affect armor. The way to a quick success is to pound an enemy with some anti-armor rounds until the plane is smoking and then finish it off with a few dum dum shells. In fact if you think of Crimson Skies as Starlancer set in the 1930s rather than as a flight sim you should have a good idea of the whole gameplay experience.
Rockets come in a variety of flavors.
The standard model is high explosives, but there are all manner of different explosive packages. From flak rockets, to chokers that starve and enemy's engine of air, or flash bombs that blinds them. It usually only takes a could of rocket strikes to down a fighter but as rockets are unguided and move slower than cannon rounds it takes some practice to use them in a dogfight. Plane construction is a really fun element of the game. Essentially it's all about fitting a selection of ready made parts together.