Cinder-block basements are a bit cheaper than poured concrete structures. This is because cinder-block laying is not skilled labor, while concrete pouring is. Hiring a concrete firm to mix, transport, pour and maintain a concrete basement will cost the homeowner or builder quite a bit. Since every aspect of the process requires skilled labor and precision labor. Cinder block work takes longer to accomplish, but it is easier.
Having a poured concrete floor can serve to anchor the entire structure. In this way, you can have a cinder block wall foundation, yet still claim to have a poured basement, since the floor is poured. The joint between the lowest row of cinder blocks and the concrete floor will be sealed, and this will create a unified structure that will keep the entire foundation from tilting in one direction or another as the house settles. This works only if you define the foundation as that part of the structure that is directly on the soil, on the footers of the foundation deep underground. You would need to eliminate the walls from your definition of foundation.
Another possibility is prefabricated cement block. In this case, the foundation can be constructed using a plastic block that is filled with concrete. The basement can either use these as cinder-block style bricks, or larger panels that are poured on site. The concrete here, since it will be poured into a mold that will serve as the outer coating of the brick, will be consistent, properly mixed and unaffected by the external environment because it is contained in a structure that is specially engineered to resist wear and moisture. The entire basement will now be unaffected by settling or curing, since the concrete dries within the block. The point here is that the builder has both a poured and block concrete basement and foundation, using the benefits of both.
If the two ideas of the basement and foundation are considered as the same, then in general, you must have one type or another. You cannot have both. Using the prefabricated block is the closest you can have to having “both” at the same time. Generally, there is not much difference between the foundation and the basement because the basement, by definition, is that room created underground from the foundation itself. In some ways, even a cinder block foundation wall is poured, since inside the blocks, the cells are filled both with rebar and concrete. Hence, the cinder block itself serves as the outer coating of the wall, not the wall itself, which is always poured.