As GFA notes, there has come to be something of a sentiment that logical consequence is a more fundamental notion than logical truth. He cites Read and Etchemendy; Dummett also takes this view (I've been reading Dummett's Frege: Philosophy of Language). GFA questions how anybody can say this when the two are (usually) equivalent; usually you can translate a logical truth into the claim that some consequence relation holds, and vice versa.
GFA does note a couple of exceptions to this equivalence. It is not exactly an exception, but is perhaps also relevant that in providing a minimal basis for a logical system, it is possible to give only rules of inference and no axioms (in fact, this is often done; the logic I'm teaching in my intro class this semester is a "natural deduction" system which uses this approach). On the other hand, it is not possible to give only axioms; some rule of inference is always needed ("axiomatic" systems normally have modus ponens, as well as some sort of substitution rule; substitution rules may be a special case, but modus ponens is clearly a rule of logical consequence). At least, the only way to give a purely axiomatic system would be to make every logical truth an axiom.
Whether all of this suffices to make consequence the more "fundamental" notion, I'm not sure. I am by nature very suspicious of claims that anything is more fundamental than anything else. On the other hand, I sympathize with some of the motives for saying that consequence is the more fundamental notion. Dummett notes that the 20th century saw quite a bit of controversy over the status of logical truths; whether they could be understood to be "analytic" (whatever that means anyway; another issue that was much fought over) and what status they did have if they couldn't be classified as analytic. Dummett seems to consider this largely ink spilled in vain (certainly nothing much was ever settled by all these debates), and also thinks there wouldn't have been so much fuss over it if people had been thinking in terms of consequences rather than logical truths. Perhaps there is more of an intuition that a logical truth needs to be about something, that something needs to make it true, than there is any corresponding intuition regarding logical consequences.
If such an intuition has indeed been a source of frivolous worries, then the equivalence of logical consequence and logical truth ought to be enough to undermine the intuition; if logical truth and logical consequence are equivalent, then it's possible, even if not compulsory, to give a reductive account of the former in terms of the latter, so intuitions that special explanations of logical truths are needed should already be undermined. But they're not precisely equivalent; as GFA's examples show, and as mine may also show, logical consequence is an ever so slightly broader notion. This surely wouldn't justify any extravagant metaphysical thesis that logical consequences are built into the structure of reality in a way that logical truths are not, but of course I don't myself think any extravagant metaphysical theses are ever justified, and if Dummett is right the great benefit of focusing on logical consequence is that such a metaphysical thesis has no intuitive appeal anyway. If we set aside such metaphysical concerns, though, we do seem to be left with a meaningful sense in which consequence is more fundamental. Still, perhaps the terminology is less than ideal, since the word "fundamental" has so many associations with the metaphysical concerns.