The quote from Livy dates to 343 B.C. when Capua appealed for aid against the Samnites (VII.31.1). In between this date and Cicero, the city was famously sacked and virtually annihilated by Rome for having gone over to Hannibal (211 B.C.).
Cicero, in the de Lege Agraria, First Oration, 1.18ff, has the following to say:
"They wish settlers to be conducted to Capua. They wish again to oppose that city to this city. They think of removing all their riches thither of transferring thither the name of the empire. That place which, because of the fertility of its lands and its abundance of every sort of production, is said to be the parent of pride and cruelty—in that our colonists [...] Our ancestors removed from Capua the magistrates, the senate, the general council, and all the ensigns of the republic, and left nothing there except the bare name of Capua;[...] the city itself might be able to afford a home to supreme power. [...] No new Rome or opposition seat of empire, will be allowed to exist while we are consuls;.
The "altera Roma" part thus appears to be a possibility which Cicero, who is against the Agrarian Law, holds up before the Senate. It is thus a rhetorical device, designed to scare. It does not necessarily reflect reality; even to Cicero, it is conditional on the agrarian law actually going through, which it didn't.
Still, Capua was prosperous enough to make the list of the top cities mentioned by Ausonius (along with Milan and Aquileia in what is now Italy). It was able to entertain famous gladiatorial schools such as that which produced Spartacus, and had fertile land, in Campania, which was probably the wealthiest district of Italy.
Considering it made Ausonius' list, as already mentioned, I only see Mediolanum and Aquileia as replacing it, though neither city was in what a Republican Roman would have considered Italy. Incidentally, Ausonius mentions Capua between Mediolanum and Aquileia, so the ranking (in Italy) would be 1. Rome, 2. Milan, 3. Capua, 4. Aquileia.
During the Republic, Corfinium, the capital of the Italians in the Social War, very briefly attained some importance, but seems to have been more of a project than reality. Possibly other cities in Campania, such as Naples, could have tried to compete; the Sicilian cities certainly, but they are - by Roman geography - Sicilian, not Italian.
Another possibility might be Ostia. If you consider it a separate city rather than an extension of Rome, it could claim the second rank. For population size, some of the cities in the Po-Valley might be considered, though I haven't really done any research to support an argument either way.
M. Caecilius M.f. Maxentius - Max C.
Qui vincit non est victor nisi victus fatetur
- Q. Ennius, Annales, Frag. XXXI, 493
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