“καὶ ἔστιν ὁ πόλεμος οὐχ ὅπλων τὸ πλέον ἀλλὰ δαπάνης.”
Why are ὅπλων and δαπάνης in the genitive case?
It’s showing a relation between the nouns - it’s not saying that war is arms or expenditure, for which the nominative would be used, but that it is, as Smyth puts it, ‘a matter of’ them.
Smyth’s “a matter of” is a bit of a non-committal translation. What uses of the genitive would he point to as similar to this? It’s hardly something that gets pulled out much in translation of a bare genitive.
My own read is that Thucydides is using this similarly to “genitive of material or contents” 1323/1324: στεφάνους ῥόδων ὄντας, ἀλλ᾽ ου᾽ χρυσίου, “…crowns being of roses, but not of gold…”.
Smyth’s “a matter of” could fit that – it’s non-committal and could fit a lot of things – and if it were me, I’d do it the way Matt says not to: “War is mostly not troops, but resources.”
Yes I’d call it a defining genitive: war doesn’t for the most part (lit. for the greater part) consist in weaponry but in financial expenditure. I think Matt put it well. The continuation explains the relationship between the two: δι’ ἣν τὰ ὅπλα ὠφελεῖ.
My own (novice!) way of understanding this usage of the Greek genitive case (and please correct me if I’m wrong!) is to remember John Steinbeck’s novel “Of Mice and Men” which is about “the experiences of George Milton and Lennie Small, two displaced migrant ranch workers, who move from place to place in California in search of new job opportunities during the Great Depression in the United States” (Wikipedia). In other words, in (somewhat archaic) English the word “of” can mean “about,” and since posession (“of-ness”) is the common meaning of the genetive case, I would read “war is not of arms…” as meaning “war is not about arms…”
“Of” in Steinbeck’s Of Mice and Men represents the well established form of Greek titles in περί (~ Latin de). Thucydides could easily have said that war is not “about” arms (περὶ ὅπλων), but interestingly he doesn’t, he just uses the bare genitive, which I take to mean “war does not consist in” arms, or as Smyth puts it, “war is not a matter of arms.” And this more nuanced phrasing contrasts also with the stark equation of subject and predicate that we find in πόλις γὰρ οὐκ οἰκήσεις ἢ γῆ ἀλλὰ ἄνδρες, discussed here earlier. (“A city is not dwellings or land but men.”) Both are rhetorically striking locutions, and both, predictably, come from speeches.
Aha, thanks! 
So what would be the grammatical term for this usage of genitive? (I mean,like we say genitive of value, genitive of possession etc.), what would this usage be?
(Apologies for the two posts below. I tried to edit my post but I accidentally clicked the “quote” button.If one of the moderators could delete it I’d appreciate it.)
I think Joel’s and Michael’s posts make good cases. I’d just say that many grammar books make the point that these categories are not exhaustive or absolute - for example, the Cambridge Grammar says that ‘the genitive is often used to express a close relationship between two nouns, without expressing the precise nature of that relationship’ (30.1, note 1), and Goodwin that the ‘classes are not exhaustive; but they will give a general idea of these relations, many of which it is difficult to classify’ (section 1085).
I’m also someone who is easily satisfied with non-committal answers, so probably not the best person to deal with questions like this!