The first sentence applies when both parties have a specific bus or bus line in mind. It is a compact way of saying: "The expected bus did not arrive today." The focus is on the lack of occurrence of something expected.
The second sentence has no limitations and so its reference is vague. It could refer to any bus on the street, in the city, or even on the planet, and so is less applicable when the behavior of a specific bus or bus line is at issue. Even if you did use it, you would be more likely to say something like: "not a single bus" to focus on the quantity.