We need money.
This seems equivalent to “we have need of money” in English, which sounds old fashioned to me (at least in English). I would have expected “wir brauchen/benötigen Geld” if the English clue is “we need money”. Are they equivalent in German? Is “wir haben Geld nötig” more old-fashioned or more formal than brauchen or benötigen?
Very difficult question.
“Wir benötigen Geld.” is more formal than "“Wir brauchen Geld.”
Maybe the versions with “Not” in it suggest some more urgency.
I do not perceive one or the other as more modern.
However, all differences are small, so this is just my personal view.
It is also worth mentioning that there is another level of nuance to the negative version of this (“es nicht nötig haben”):
It expresses sovereignty or supremacy:
Ich habe es nicht nötig, mich von Ihnen beleidigen zu lassen.
I don’t need to be insulted by you.
I.e. You can’t hurt me, I’m above these things. You are not worth my time.
Wir haben es nicht nötig, unser Haus zu putzen.
We don’t need to clean our house.
I.e. we’re rich enough to afford a housekeeper.
Likewise, the positive version can also pejoratively express an exaggerated, unnecessary need (if one says it to others):
Ich muss mal wieder ins Bordell. - Na, du hast es ja nötig.
Du hast die ganze Schokolade auf einmal gegessen? Na, du hast es ja nötig.
So the bottom line: this expression does indeed seem to imply more urgency than a simple “brauchen” or “benötigen”.