Typically there was continuation, but this is a good question!

Let's start with Egypt: the Egyptians in the Iron Age were quite like those of the Bronze. They spoke Egyptian, wrote Egyptian, had Egyptian beliefs, dressed like Egyptians, but their military had changed. Whereas Bronze Age Egypt had been famous for charioteers and specialized archers, the Iron Age would bring in short straight swords, less prominence of the upper warrior caste, and generally what could be called a democratization in that what had once been very organized and structured society was now recovering. The kingdom wasn't as strong, and it would soon fall to the Greeks. Ptolemies tried to merge Greek culture with Egyptian, though they didn't like each other that much as a general rule.

Mesopotamia: Despite what seems like a big difference, there isn't that much change here. Bronze Age Mesopotamia is noted for Akkadian literature and its classical religion, similarity to Sumerians, and so on. The Akkadians of the Iron Age had similar military structures, spoke Aramaic (for the use of Akkadian had been purely literary for quite a while), and acted much like their predecessors. They weren't really as affected by the collapse.

Levant: Canaan underwent significant change. In the wake of the collapse spawned the centralization of the tribal confederation of "Israelites", the movement away from nomadism and seminomadism for form a centralized monarchy. The invading Philistines, who archaeology would indicate may have been Greek originally, presented a threat to the Israelites that allowed for greater national unity to survive the troubles. The Bible details, some parts history and some parts legend, the conflict between these groups. Other Canaanite groups like Ugarit had been collapsed, with unfinished tablets left in the ruins of the city alongside piles of bones and arrowheads. Phoenicia had fallen drastically from prominence, replaced by their daughter-state of Carthage. Israelite accounts seem to indicate that weapons were hard to come by during the early millennium, in the transition from bronze to iron. Syria, meanwhile, had changed quite a lot. Once part of the vast Hittite Empire, Syria had collapsed into a series of states where Hittites blended with Arameans, becoming more Semitic in the process. The area of Jordan didn't change much, once being Midian and now Nabataea, both Bedouin confederations.

Greece: This is where you might find civilization changing the most. While Bronze-age Greeks bore a fair deal of similarity to their Minoan neighbors (and presumably the Pelasgians who lived there before, likely connected to the Minoans), the later Iron Age Greeks (ironically the ones we think about more typically, with the big bronze shields) were significantly more distinct. For a contrast of how this applies in a sense beyond just clothing and architecture, compare the culture of Macedon to that of Athens and Sparta.

Italy: The Romans tended to be considered a continuation of the Etruscans. Said Etruscans did not even consider them a separate people. Romans used similar symbolism (fasces, for instance, for proto-fascism), similar religions, similar architecture, and it was once said that there was but a single difference: "Where the Etruscans say clouds come together that lightning may occur, we Romans say that lightning occurs when clouds come together"