The short answer: cavalry.

At Cannae, the Romans had a double sized field army of around 80,000 men but only 6,000 cavalry. In a recent defeat, the Roman centre had managed to punch their way through the Punic centre and escape. So the idea was to deploy their legions much deeper and use their added momentum to punch through and defeat the Carthaginians before their wings collapsed.

Hannibal on his side had only half the amount of infantry (40,000) but 10,000 cavalry. He enacted a plan to suck in the Romans and envelop them. He placed his more expendable Gallic troops in the centre and his veteran Africans on the right and left in column. He was gambling that his centre would hold out just long enough for his African soldiers to attack from the vulnerable flanks. Plus his majority in cavalry would allow give him control of the battlefield.

And things appear to happen pretty much to Hannibal's plan. The outnumbered Roman cavalry was routed and the Carthaginian cavalry then hit the Romans from the rear while they were attacked in the flanks by the Africans. A few did break through the centre but the majority of Romans got pushed in and compacted so tightly that it impeded their ability to fight.

Fast forward to Zama and Hannibal probably had another plan, to deploy his army in 3 lines with the most expendable in the first. His plan is to have the Romans wear themselves out on his first few lines and then rout them with his 3rd line of veterans from Italy. In addition he had 80 elephants which could do considerable damage.

But Scipio had planned for this and arranged his army not in the usual checkerboard pattern but with lanes to channel the elephants through. Some were also routed by skirmishers and back into their own forces. Roman and Numidian cavalry used this confusion to rout the Carthaginian horse and drive them from the field.

Hannibal's plan fell apart when the Romans managed to halt their advance after defeating the first two Punic lines and reform. This meant that Hannibal wasn't going to fight a tired Roman line that's slightly disordered from chasing the routing Carthaginian 1st and 2nd lines - but a reformed and reinforced one. Both sides may have extended their lines, the Romans pulling up the Principes and the Carthaginians using the rallied survivors of the first two lines. But all for naught as the returning Roman cavalry hit the Carthaginians from behind and routed them.