1) But "the plural-only noun "де́ньги" technically IS the plural of "the colloquial singular noun "деньга́". So I guess your question here makes little sense, I can see no dichotomy here. It is plural instrumental form of the same word either way.
Yeah, the plural "деньги" gained additional generic sense that "деньга" arguably lacks - "деньги" can refer to a bank account or a BitCoin wallet, while "деньга" refers to physical cash, to a coin or a banknote. Still does it make them really different words? I personally do not think so.
Compare with plural "люди" gaining the additional meaning of singular "people" and remaining in modern use due to that generic sense, while singular "людь" falling out of usage together with "полюдье" and other feudal-era terms.
Even if you consider them semantically being different words, homonyms, they still technically are merely forms of one word. So, as for me it is exactly the case of "ultimately amount to the same thing". At least when the question is formulated in technical terms (from which... stems).
2) Now if you ask what was the specific meaning of the word "деньгами" in that phrase, then we can put it to the test using synonyms. Granted, this test perhaps is not easy for foreign person living outside "language native environment". However, I would try to bring some examples. We try to replace the word in question with other words, that convey only or mostly one of those meanings we try to differentiate.
Купюра - (here) banknote. It does not have a meaning of "money in general". So "у нас туго с купюрами" would refer to paper cash specifically. Does it sound idiomatic? I'd say no. It can be used in a professional environment, where a cashier or ATM maintainer may report an abnormal shortage of paper cash to one's boss. But you would next to never meet it in casual life.
Банкнота - banknote. Same as above.
Лавэ - that is a rather vulgar semi-bandito word for "money" (plural деньги). "У нас туго с лавэ" - it sounds perfectly natural to me (with that gangsta accent, which is not always welcome, but that is another issue).
Монета - coin. This is a bit tricky. Cause in plural "монеты"/"coins" MIGHT be used colloquially to refer money-in-general, quite like plural of "деньга" does normally.
My guess (unchecked) is that деньга/таньга was originally a Tartar word for coin. And if that word for coin gained an extra generic meaning in plural, then it is only natural for another word denoting coin to gave the same trait.
So, back to the case, in the aforementioned professional environment "у нас туго с монетами" would refer specifically to lack of coins (while paper cash might be in abundance). However in general population "у нас туго с монетами" would be seen as a somewhat vulgar wannabe-gangsta way to say "short of (in general) money".
By default, w/o context, деньги would refer to money in general and монеты to coins, the particular physical form of money.
However when pressed by the "у нас туго с ..." context the "монеты" would be normally pushed to its secondary more vague and abstract meaning.
Рубль - rouble. Might refer to the currency (roubles vs dollars vs pounds vs ...) or to a cash of the specific value (coins or banknotes bearing value of exactly 1 rouble, not 2 or 10 or half). Here again "у нас туго с рублями" would normally imply that professional environment situation - or some pretentious person intentionally garbling the language for the sake of avoiding normative "туго с деньгами".
Наличность - cash. Singular-only. "У нас туго с наличностью" would imply "short of pocket money" kind of situation, short of money that could be handed over right this instant. Technically "наличность" is short for "наличные деньги" - "cash money", however "наличный" in general means some resource "instantly available, existing here and now". So that phrase would imply that in general there are some valuables (like bank account to draw cash from, or like expensive property to be firesold) that can be converted into the required cash, just it would be troubling and bothersome, so the doubt is conveyed whether this route to be taken or the whole "deal" refused.
ATM I can not come with more synonyms to check and compare.