I'm with those who've already stated that Greece needs to be conquered before an attack on Carthage could be made. I don' think it matter whether its the first (Marathon) or second attempt (the Hot Gates).
If the Persians win in the second attempt, and many of the Greeks flee west, to Sicily and Southern Italy, then the balance of power in the western med is going to change. The Carthagians are suddenly facing a much stronger Greek presence, one that is probably also going to cause some pretty major political changes in the Italian and Sicilian Greek city-states.
So instead of Rome uniting Italy and then facing off against Carthage, the Greek refugees come to Italy, and with the shared experience of the "Long Trek" (or whatever you want to name it) unite the Italian and Sicilian Greek City-states into a new federation, one which adds the greekophile city-states of central Italy as allies (this includes Rome). Control of Sicily is a major issue for both Carthage and the Greeks, and war soon ensues.
In the ensuing face off between Carthage and the (Hesperian?) Greeks the Persians would definitely be interested in getting involved. Since the Greeks are probably being a REALLY big pain in the side of the Persian Empire (I'm thinking constant support to remaining Greeks in rebellions, state-sponsored 'piracy' in support of the rebellions), the Persians could offer some help to Carthage, of what kind I'm not entirely sure. I would imagine that it would be in form of at least a naval variety, so the Carthagians get support from their Punic cousins in the Old Country.
The struggle between the Greeks and Carthagians ends up as the death struggle that it was between Rome and Carthage. In the end the Greeks are able to overcome the Carthagians, in the process adding Spain and Sicily to its dominions. The Sack of Carthage after the Second Punic War ends the Carthagians presence in the West as an independent people.
In the process of winning against the Carthagians the Greeks would have gained enough strength to really challenge the Persians. The Greeks liberate Greece, and land troops in Egypt in support of a local rebellion there. The decaying Persian Empire is soon in retreat, pushed behind the Zagros Mnts. before the Greeks finally stop.