What might such an "Old Roman" republic look like? And how might they hype up their non-Greek aspects considering the Romans already shared so much with the Greeks like a majority of their religion?
That's hard to say. The obvious answer is "start with imagining Rome as it was up to about 300 BC and extrapolate from there", but that tells you quite little. This is what Cato had in mind, envisioning a quasi-romantic past under aristocratic rule. In reality, the Conflict of the Orders is a done deal, and Rome has changed from its earlier form. Military reforms are also a given, since they were devised in part for a conflict against the Greeks. Even if said conflict ends in defeat for Rome, that still won't negate the reforms.
The actual differences compared to OTL will be in Rome's attitude to the outside world, and in culture. If Rome's expansion is broken, it can't just be re-directed. The ascent of an empire is based on the exploitation of subdued territories, which will finance new expansion, while the territories captured earlier are slowly integrated (romanised). If that process is halted to a significant extent, the wealth pump runs dry, and the imperial ambition becomes impossible to finance. Thus: Rome ends up as a nation instead of an Empire. That means less comopolitanism, less religious syncretism etc. -- Less change. More of a fixed national identity. This also means that the imperial way of financing the government will be impossible, which means more of a tax burden on citizens. Which means, ultimately, more influence for those who pay taxes. Which means a path to political and social status for wealthy men of lowly origins. Moreso and earlier than in OTL. (That's
not something Cato would've expected!)
Back to culture. In OTL, the Romans increasingly patterned their writings, poetry, visual art, architecture and philosophy on the Greek example. Won't happen here. No
Iliad. No other poems clearly based on Greek examples. No increased tendency to make monumental architecture look like the stuff in Athens. No styles in statues and murals copied directly from the Greeks. As far as this stuff is concerned, you
can look at Roman examples from an earlier date, and expect the ATL stuff to look more like that than like the later stuff from OTL. As far as philosophy is concerned: the Romans imported Greek philosophy wholesale and really went to town on it. I vaguely suspect that in this ATL, based on earlier Roman attitudes, Rome would be somewhat "un-philosophical". They'd pride themselves -- like the Spartans (not that they'd appreciate the comparison in this ATL!) -- on being above endless meandering speculation. Romans are Men Of Action!
Real men! Not like those effeminate Greeks! (Okay, that's obviously a caricature, but you get the point.)
Rome held Punic world. But although some Punic cities like Utica saved themselves from the kind of destruction inflicted on Carthage, they remained culturally uninfluential.
What would be needed for Romans to treat the Greek cities they spare like they treated the Punic cities they spared? Both Carthage and Corinth were destroyed the same year.
I think a major issue here is that the Roman and the Carthaginian attitudes never really
jived, so to speak. They were strategically allied at one stage, but they never realy understood or appreaciated each other's cultural priorities and mind-set. Romans conquered Megale Hellas and saw something they wanted to make their own. They looked across the sea at Carthage and the other city-states, and they only saw an inherently foreign enemy. Same was true the other way: I think the only notable Carthaginian who truly understood the Roman mind-set was Hannibal. Judging by how they approached the war, his peers certainly didn't... and they didn't really understand Hannibal, either. (If Hannibal had been born a Roman, he'd have been a great one.)
The Romans saw Greece as dazzling, and wanted to possess it. They saw Carthage as alien, and wanted to crush it. The two attitudes cannot be equated, because to the Romans, Greece and Carthage couldn't be equated.