I know a little bit about Assyrian iron usage, but I can't frame any of it in answer to this question. To know how much of an advantage it conferred, you'd have to first know iron's material benefits over copper-alloy metals, secondly the relative distribution and use of iron amongst Assyria's enemies, and thirdly what the aftermath of Assyrian battles was like (casualty rates, types of wounds, etc.).

I don't know any of this, so instead I'm going to pretend you asked what role iron played in Assyrian warfare, and go from there.

Iron-working appears early in Assyrian history, in the form of meteoric metals limited mainly to ceremonial daggers and prestige items. By the end of the second millennium BCE, iron was becoming more and more common as metallurgical processes for the extraction of iron from ore were developed. The resultant wrought iron was brittle, lacked the luster of other valuable metals, and was highly susceptible to rust and corrosion, but it was still regarded as a high-value commodity due to its utility and the relative difficulty in obtaining it. By the height of Neo-Assyrian military power in the 8th and 9th centuries BCE, textual references to iron are ubiquitous. It's listed in bulk amounts as an item of loot or tribute taken from the subjects of conquest (mainly in the northeastern corner of the Mediterranean region, around southern Anatolia and the Levant), and iron daggers (patri parzilli) had gone from being rare symbols of wealth and power to becoming a widespread part of the professional soldier's kit. They were important enough to become synonymous with the ferocity of Assyrian might, invoked in prose, prayer, and simile.

Daggers may have been the most literarily pleasing use of iron in warfare, but of course they were far from being the only ferrous weaponry. Archaeological evidence attests to axes, short swords, spear- and lance-tips, arrowheads (though of the latter, about 50% were bronze throughout the Neo-Assyrian period), and copious amounts of scale armor, like that which is famously depicted in Assyrian murals and friezes. We've even found plates of armor apparently meant to adorn and protect horses in the cavalry!

My source for most of this is this article, worth a read if you have the time.

Also, wasn't a dagger one of the symbols of Assur? As well as an earlier Sumerian god? I can't recall for sure, I'll look it up when I get back to my books. Hope any of this was useful!