Obviously the Greek and later Roman world made enormous contributions to what we would come to call "western civilization", but had the Persians conquered those parts of Europe, would the world really have been worse off?
From what I've heard slavery was largely outlawed in the Persian Empire, and the empire in general was more egalitarian than Spartan and Greek society. Is there any truth to this? If slavery was outlawed in Persia, why do stories like 300 portray Xerxes army consisting almost entirely of slaves?
It is not accurate, on at least two important counts.
How much did the flowering of international engineering, scientific, and scholarly endeavors in the Hellenistic Period depend on the nationality of who was in charge? Probably a fair bit, but we can never be certain: Hellenistic Egypt was the fountainhead of much of this intellectual flowering, but remember Egypt had been under Persian control before it was under Macedonian control.
To what extent was the intellectual flowering in the Hellenistic and Roman periods responsible for the developments during the Renaissance that led to the Industrial Revolution? It certainly helped: Greek mathematical texts and Ptolemy's geographical work were nothing to be sneezed at. But western Europe had been getting on very nicely since the Black Death, and if Ptolemy and Euclid hadn't made in into the west, I personally doubt that much would have been very different. We might not have Homer, but we would still have western mediaeval intellectual and technological traditions, and they are certainly nothing to be sneezed at.
However, with the benefit of hindsight it is easy to see that Persia could never have held all of Greece. The problem of maintaining supply lines around the edge of a sea that was completely dominated by the opposing force, the fact that Greek hoplites were just better armed, the fact that there had been an emerging Panhellenic ethnic identity there for at least a couple of hundred years, and the fact that the Greeks were very well positioned economically (they had currency!), in combination made the conquest of Greece a much, much bigger problem than the Persians could ever have anticipated. The fact that Sparta existed made the rout of the Persians faster than it might otherwise have been; but if Sparta hadn't existed, Argos might well have felt a bit more of a sense of responsibility to do something about the invaders.
If slavery was outlawed in Persia, why do stories like 300 portray Xerxes army consisting almost entirely of slaves?
The author of 300 completely bought the Greek line of thinking that saw the Persians as one free man ruling over a nation of slaves.
The constitutional side of the conflict between Greece and Persia was not freedom versus slavery, but more like republicanism versus monarchy. Athens and Sparta liked to play up the differences between their constitutions, but from an outside perspective they were much more like one another than either of them was like a monarchy. The difference wasn't about whether everyone was enfranchised ("democracy") or only an elite ("oligarchy"), but a matter of degree: exactly how small a proportion of the populace belonged to the elite? In both Athens and Sparta, slaves/helots made up the vast majority of the population. There was nothing "free" about either of them unless you were a member of the elite. It just happens that in Athens, the elite was a significantly larger body than in Sparta, and included non-urban ranchers/farmers, so they called their version of oligarchy by a different name.
And by the way, the story of 300 is pretty much irrelevant to the repulsion of the Persian invasion. The Thespiaeans, Spartans, and Thebans lost the battle, and it achieved essentially nothing. The battle only became important in later years, as part of Sparta's propaganda campaign to cast themselves as the heroes of Greece and achieve hegemony over other states, at the expense of their main competitor Athens. Athens was in control of the Panhellenic alliance, or Delian League as it became known, that repelled the Persians from the east Aegean from 477 BC onwards; as soon as Athens gained control of the allied army, Sparta threw its toys out of the pram and abandoned the alliance.
The very idea that that was a pivotal moment in western culture its more a pop-culture thing than a real thing, much like the Dark Ages being devoid of any scientific or cultural developments worth mentioning. Alexander's conquest of the Persian empire and most of its possessions was far more important for the development of western culture than the battle of Thermopylae.