Kraftstoff/Treibstoff are general terms for something that powers something.
There are many types of Kraftstoff/Treibstoff and you're not limited to cars/trucks if you use it. You can also use it for planes (Flugzeugkraftstoff => Kerosin) or even rockets (Raketentreibstoff). It's a general term that is not old! A typical place where you find it would be the fact-sheet at the car dealer as:
If you talk about cars or trucks, Benzin and Diesel are the traditional forms of Kraftstoff, the former being split into sub-types by the octane number ("Normal 95", "Super 95", "Super Plus 98", etc.). The latter can be split into types as well, like "Sommerdiesel", "Winterdiesel", "Bio-Diesel", etc. There are other Kraftstoffe, however, like Erdgas or LPG (Liquid Power Gas).
Sprit appears to be a rather colloquial term for everything that drives the engine of a car (people use it for alcohol they drink as well, which is definitely colloquial), but it is actually used in non-collquial environments, for example papers or the news:
Sprit wird wieder teuerer.
People would know that you're talking about Benzin, Diesel, Erdgas and LPG together.
The way people refer to "fuel" depends on what they want to say. There are many options. There are a few rules you can stick to:
If we talk about our own car, we usually refer to the real type of fuel our car needs, so we wouldn't generally say Benzin if our car needed Diesel. Or we would use Sprit. Or we would avoid the term at all. Examples:
Ich muss ja schon wieder tanken!
Der Sprit/das Benzin/der Diesel/der Tank ist schon wieder leer!
If you talk to people about the "Spritpreis", you could say:
Benzin wird auch wieder teurer!
Most people would know the same applies to Diesel then.