The answer to what the world would have been like without the Roman Empire depends entirely on the reason why there would not have been a Roman Empire in the first place.
There are a couple of plausible scenarios.
1. Brennus burns Rome and kills every last man woman and child, sows the fields with salt, and eats all the dogs. After that it's hard to say: the urban societies of the Etruscans were already somewhat in decline by this point, and I can't buy a Samnite or Sabine empire. Possibly without a strong Italian power, Greek influence would have spread up the peninsula.
2. Alexander didn't die in 323 BC, but lived at least another 20 years, and carried out what were reputed to have been his plans for western conquests against Carthage and Sicily. I find it likely that he would have been drawn into the Italian peninsula, and given for the sake of argument that he would have won, that's Rome nipped in the bud. After that, what happens depends on whether Alex ever stopped to consolidate his empire and provide for an orderly succession - assuming that he left an adult male heir, we can posit an Alexandrine Empire lasting for a while, I think. Longer if the Greeks adopted some Roman civic virtues. This scenario is put forward in Mary Renault's The Nature of Alexander the Great.
The successful conquest of Rome by Phyrrus is an alternate version of this scenario, with smaller results - Phyrrus conquers Italy, Sicily and Greece.
3. Rome loses to Carthage in either of the wars. I can't bring myself to believe that Carthage would have eradicated Rome or replaced the population, so the eventual outcome of this is harder to imagine - it possibly might have just meant a delay in events, or a later repeat of scenario #1 - no strong regional power in Italy.
While I would never say that history is inevitable, there are important reasons why Rome succeeded and Carthage, the Greeks and the Celts did not - and just removing Rome via deus ex machina wouldn't necessarily change those conditions. The Greeks had much more conservative and parochial notions of citizenship than the Romans did (as did the Carthaginians), something that limited the ability of both to sustain expansion over time. Geographical position is also very important.
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