As you already admit yourself in the question, you're asking about 2 very different things: the world of Homer and Bronze Age Greece. It's important to keep the two completely separate. The Homeric epics contain some references to the Greek world in the Late Bronze Age (c. 1250-1100 BC), but the Greeks of the time when it was composed (c. 700 BC) did not have a memory of such an age and did not understand the technological implications of setting a story in that age. Instead, the Homeric epics are set in the Age of Heroes, which is emphatically not a time of normal humans and human history. Its warfare is most likely inspired by the contemporary warfare of the early Archaic period, when both bronze and iron were in general use.

The heroes of Homeric epic all carry more or less the same bronze weapons and heavy armour: helmets, breastplates, girdles and greaves. With a few exceptions (such as Aias' famous tower shield made of nine layers of oxhide) they are not distinguished from each other by the nature of their equipment, but by its value. There is a tremendous emphasis on how nice people's gear looks and how much it might be worth as an item of exchange. Some armour is made of linen or bronze while other pieces made of gold or tin are clearly fantasy items meant to show off the heroes' wealth. The point of all this is to show that this world is full of supernatural heroes with supernaturally deep coffers. Their way of fighting is more or less the same for everyone.

While the great mass of warriors in the Iliad would not have been able to afford much or any of this armour, they also rarely make an appearance in descriptions of heroic deeds. Heroes cleaving their way through hordes of enemies aren't described as fighting these poorer masses, because there was no glory in that; to win fame you had to kill someone famous. The ones killed by Agamemnon or Sarpedon or Achilles are all named lords and we can assume that they would have all worn similar types of armour. Many of the implied victims of such aristeia (Great Deeds) may have been the unlucky infantry meant to back up the lords in battle, but the only opponents worth mentioning were other heroes, and these would not have been at any disadvantage of equipment. There is no better proof of this than the scene where the Greek Diomedes and the Trojan Glaukos exchange armour as a courtesy when they learn that their ancestors were guest-friends. When heroes like Hektor carve a bloody path through the enemy, it is because they are better than other warriors (or because the gods are helping them), not because they happen to have a sharper spear.

So much for the Iliad, where technological differences clearly cannot and do not explain any part of the story. But what about the actual Bronze Age?

Unfortunately there are no surviving descriptions of fighting in Bronze Age Greece. There are some neat images, like the Mycenaean sword inlaid with a lion hunting scene that shows the detail of warriors' equipment (and also, you know, is a sword), and the so-called Combat Agate found in a grave at Pylos, which shows two warriors fighting. What these images tell us is that the use of metal-tipped spears and metal swords was well-established, along with massive figure-of-eight tower shields (presumably of reinforced hide, though wood or wicker are other possibilities) and crested metal or ivory helmets for personal protection. Even if we assume that only the wealthiest warriors would have been able to afford this gear, it was at least common enough not to raise any eyebrows, and the enemy would no doubt have bronze-armed warriors of their own to neutralise these champions.

The reason for this is that equipment like this tends to spread fast, not just because it works, but because it is treasured as a status symbol. In the ancient world, new military equipment and tactics often spread not through conquest and adaptation, but through what's known as peer-polity interaction - a fancy word for "if the guys cross the bay have it, we should have it too." Prized metal items of all kinds are susceptible to this, as are forms of military organisation, military architecture and the like. They were not kept secret and only discovered at the next battle; they would be shown/built/worn long before they were ever used in anger. And no lord wanted to live in a village if his peers were living in fortified palaces.

But even if that kind of imitation wasn't a factor - even if we imagine the first battle in which the first man ever used a bronze weapon - just having a slightly more effective weapon or piece of armour doesn't make you Superman. A rock to the head will take you out, whether you're carrying a stick or sword or an AR-15. Stone clubs, axes and spears, and arrows made of flint or bone, don't instantly become useless the moment someone introduces copper or (eventually) bronze. Even covered head to toe in bronze armour like the massive Mycenaean Dendra panoply, you are not completely invulnerable. Indeed, the humble rock never disappears from ancient warfare; in the hands of slingers and articllery crew it remained one of the most lethal weapons around, but even in the hands of a poor and desperate warrior with nothing but his own muscles to rely on, it can lay a guy out flat if it hits. No ancient warrior would think, just because he was holding a new kind of weapon, that he was better than all the other warriors in the world.

This is a very general question, but since you directly reference the Trojan War, let me latch onto that for a moment. First of all, the fact that bronze is used in the Iliad means very little. Hans van Wees, in his article "The Homeric way of war: the Iliad and the hoplite phalanx" (Greece & Rome 41, p. 134), makes the point that the material of choice for the manufacture of weapons and armour in the heroic world is bronze, no doubt picked more for its lustrous appearance than its superiority over iron. Homer mentions iron a few times and also refers to the forging of weapons, so he's quite well aware of the material.

When it comes to the actual Bronze Age, as in the period in the Aegean preceding the Iron Age (before ca. 1000 BC), the typical material used for weapons (and armour) is bronze, but stone arrowheads do remain in use for a long time (e.g. flint, obsidian). In her unpublished PhD thesis, Elements of Mycenaean Warfare (1990), Diane Fortenberry writes that "Stone arrowheads continued in use to the end of the Bronze Age, though their numbers in comparison to bronze points decrease as time goes on" (p. 226). Slingers, of course, were also active in the Bronze Age and they used stones as ammunition. As Fortenberry also makes clear, and as a reading of the Iliad illustrates, stones would also be picked up and thrown by hand, and rocks could be thrown at assailants by defenders at e.g. the Lion Gate in Mycenae.

There were, of course, differences in equipment between different warriors, but there is no evidence to suggests that only "elite" warriors used bronze weapons and everyone else stone weapons. Swords are only known in bronze, and spearheads of the types used in the Bronze Age were also always made of bronze. Armour was made of bronze, too, like the famous panoply from Dendra, dated to the later 15th century BC, though some have suggested leather or linen may have been used as protection for the body as well: see again the relevant discussion in Fortenberry, or the overview in Tim Everson's Warfare in Ancient Greece (2004) or my own Henchmen of Ares: Warriors and Warfare in Early Greece (2013), based on my PhD thesis.