Well, here's a probably oft-repeated topic, but what if Carthage does better in at least one Punic War, and manages to survive against Rome?
Well...
-1st Punic War: the only way to win is by countering Roman ship-boarding strategy...how about a thin plate of metal being attached to the ships to prevent the Romans use their corvus...? (just like in robertp's Egyptian and Hittite TLs)
-2nd Punic War: Hannibal was nearly win the war...he just need some siege engineers...
The other way was to ensuring the royalty of Numidians to Carthage, defeat the Romans at Zama, and re-invade Italy... (with siege engineers, of course)
-3rd Punic War: there's no way, I think...Carthage was only a city-state without any territorial holdings outside OTL Tunisia...
So, if Carthage win in 1st Punic War, I could imagine the continued hostilities between Rome and Carthage...but if Carthage win the 2nd Punic War, and Hannibal successfully razing Rome to the ground...that would be a long ATL story...
Oh, and welcome to the board...