These are interesting questions. I've tried to offer answers below, but we have to answer another question first:

What is a Greek phalanx?

If we want to know how important or how revolutionary the phalanx was, we first have to understand its nature. On the face of it, this nature is pretty simple: the formation that eventually became known as the phalanx was a group of heavily armed infantrymen arranged in a grid pattern of ranks and files, whose battlefield function was to engage the enemy in close combat. That's pretty much all there is to it.

A lot of modern scholars have tried to polish this basic idea to a high shine, trying to make the phalanx out to be something it never was. They will claim that the formation was a rigorously drilled, carefully orchestrated, moving wall of spears and shields, a steamroller of bronze-clad men whose very appearance struck fear into the heart of its enemies.

This all sounds really cool, but there's not much evidence to support it. Throughout the Classical period, the phalanx remained untrained; it may have been precisely drawn up, but its charge into battle turned it into little more than a screaming mob of heavily armed men. Of the men who made up that mob, few will have been able to afford the full bronze panoply, and even those who could seem to have gradually abandoned the hot and heavy equipment we associate with hoplites. By the late fifth century BC, a hoplite was simply a citizen armed with spear and shield, and few wore more armour than that.

This fairly primitive picture doesn't just change the way we imagine Greek battles. It should also make us question just how special the Greeks were when they came up with the concept of the phalanx. They were hardly the first to come up with the idea of massing together heavy infantry for effective close combat; there are reliefs showing Egyptians, Akkadians and Assyrians doing exactly the same thing, centuries before the first hint of a phalanx formation in Greece. The Classical Greeks even acknowledged this, referring to Assyrian and Egyptian infantry as "hoplites" and describing their heavy equipment in some detail. Xenophon recognised that the Egyptians were probably more tactically capable as heavy infantry than the Greeks were.

Really, the development of phalanx-like formations is just a logical result of an increase in the use of close combat weaponry paired with an increase in state organisation. It can happen in any society where these factors are present. In Greece, compared to its Eastern neighbours, it happened pretty late, but because they wrote and drew and sculpted us such a nice image of it, we somehow associate it more with them than with other peoples of the Levant. And this brings us neatly to your questions:

Why was the Greek phalanx so important?

In the grand narrative of military history, as I've just explained, it wasn't. However, the phalanx is a sign of growing state control and military organisation, and as such, it can serve as a marker of how Greek civilisation was advancing to catch up with the empires of the East. Effectively, the appearance of the phalanx is proof that organised warfare was moving west. This is probably the first time it ever did so.

It was a timely development, too, in that the phalanx (and the trireme, another invention borrowed from the East) allowed the Greek city-states to defeat the Persians in battle and carve out their own sphere of influence on the edge of the civilised world. Since the Greeks appear to have specialised in heavy infantry combat from the Persian Wars onward, Greek mercenaries became a desirable commodity. Meanwhile, the Greeks' own incessant wars made them increasingly skilled at organising, financing and leading armies. (It's worth noting that the word "phalanx" as a technical military term dates to the fourth century BC - representing the formal conceptualisation of a practice that had been around for over a century.) The culmination of all this development was the army reform of Philip II of Macedon, who created a standing army built around a phalanx of pikemen that would go on to conquer the Persian empire.

The phalanx, then, was not really a breakthrough in itself, but a sign of the increasing sophistication of Greek warfare, that eventually allowed them first to check and then to compete with the great empires of the East (which had traditionally been their superiors in military organisation and tactics).

However, when confronted with more advanced infantry formations, such as the Macedonian pike phalanx or the Roman manipular legion (or even the Spartan phalanx), the hoplite phalanx almost invariably crumbled. It needed better drill, a better command structure and better individual training and discipline to stand against other organised heavy infantry.

How has it influenced warfare today?

There's a strange school of thought led by Victor Davis Hanson that claims that the Greeks stood at the cradle of a continuous Western tradition of deciding wars by decisive battle between heavy infantry on open ground. In their view, the phalanx and its fighting style continue to influence our thinking to this day. However, this view is widely discredited now - I wrote a post about its status among Greek Warfare scholars here. The Greeks didn't really always fight this way; the Persians, on the other hand, did. If anything, the notion of pitched battle as a particular military "culture" only confirms the idea that the Greeks borrowed their advanced military organisation from the East, which is where the origins of this supposedly "Western" ideal must be sought.

Other than that, there really is no direct line between Greek warfare and modern military doctrine, except insofar as the Greeks pioneered the study of military concepts that retain their validity to this day. This has little to do with the phalanx, though, and more with the development of warfare as a technê, which might be something for another thread.

I hope this answers your questions! Feel free to ask follow-up questions/demand to see my sources/yell at me for my revisionist delusions