Grice was writing in the 1950s and '60s and he was a philosopher, not a linguist. Thus his views of "what is said" are not linguistic ones, and indeed they're not all that clear to his readers.
There is, in fact, a problem with talking about "what is said" that emerges as soon as the description goes beyond the sounds uttered, because anything beyond the physical and physiological data require human interpretation, and that varies. Grice's intuition, I think, was that while but is logically equivalent to and, there is something more involved when you say but, and it refers to the emotion associated with the clause introduced by but.
When one says He was a politician, but he was honest, one signals some surprise, as if that combination of propositions were not something one would expect. However, it's not part of the logical meaning, because whatever that "something more involved" is when we say but, it has no effect on the truth of the two propositions. That is, in both
- That company makes bagels, and they're not kosher.
- That company makes bagels, but they're not kosher.
it is true that that company makes bagels, and that the bagels are not kosher. That is to say, logically they are identical. The difference is the presupposition, which accompanies but, to the effect that bagels are, or ought to be, or are expected to be, kosher.
In the decades following Grice, the concept of Presupposition has emerged to deal with the problems of but, as well as a host of other semantic/pragmatic problems. Presuppositions are considered to be a part of pragmatics, while logic is considered semantics.
In philosophy, "semantics" is usually limited to logic; hence the motto Semantics underspecifies, which refers to all the things that logic doesn't deal with.