Carthage drew a lot of its wealth from its agricultural and maritime activities. As a result I don't think they had quite the manpower base of citizen farmers to draw on for a levy army to the degree Rome did. This perhaps made it harder to thoroughly dominate neighbors and keep them in the empire. Instead they had to deal with growth a bit differently.

As Carthage expanded they would project their economic and military influence around North Africa. It seems that they remained content with having the subjugated neighbors give tribute in money and soldiers. In future wars Carthage could use these troops to fill the army plus use the cash to hire mercenaries. But in times of peace they could instead use the funds for other measures. I assume this was more advantageous for the mercantilistic minded Carthaginians.

So though we might think of the Punic forces as being largely mercenaries, in fact the troops were often coming from subject nations. Why Carthage chose to rule in this way rather than the way Rome did, I'm not quite sure. But one could probably trace some of it back to Carthaginian roots as a colony with interests primarily in controlling the seas.

I'd recommend the book "The Punic Wars" by Adrian Goldsworthy which has a good deal about Carthage. The early history of this north African power however is a bit harder to come by so answering your question might be hard.

I think that there's something to be said for the mentality of Carthaginians that made them keep a cultural distance from their subjects rather than assimilating them as Rome liked to do.

Carthaginians were essentially outsiders in North Africa; they had been there for a long time and had mixed somewhat with local North Africans - the worship of Tanit is likely a result of this cultural exchange - but they were still quite Phoenician culturally and, considering the mindset of the era, would have seen the North African and Iberian peoples they subdued as uncivilised and ripe for exploitation: tribute, heavy taxation, soldiers and the like. Raising them to the level of Carthaginian citizens in terms of social status would be hard to conceive considering the oligarchic nature of Carthaginian government.

Furthermore, assimilation in the fashion of the Romans wasn't common practice. Empires at the time tended not to operate that way. The great empires up to then - like the Neo-Assyrian empire, the Achaemenid Empire or the Successor Kingdoms - did not culturally assimilate. For the Successors this was frequently an issue as Greek kings were ruling over non-Greek peoples. Their armies too included many "foreign" units, which could be called mercenaries much like Carthage's armies.

When not recruiting subject peoples, more conventional mercenaries have their advantages, too: they are closer to professional soldiers in a world where many states' armies were levied from the populace. And seeing as Carthage was very rich, they would not have typically had cause to worry over high expenses.

As you say, this changed after the First Punic War, when the economy was in poor shape and all the mercenaries had to be paid. I suppose the reason they didn't change their approach by the Second War comes down to not having much time between wars to consider and then engage in a complex, colossal change like cultural assimilation (it's easier, after all, to say that assimilation was a better path in hindsight) and the fact that Hannibal and his clan, who provided the majority of Punic military force at that time, weren't operating under the conventional Punic system anyway. The army of Hannibal was strongly tied to him and his family since many had served Hannibal and even Hamilcar before him. They weren't thrown together once the war started and given a commander like in the First War.

Add to that the fact that communication between the "Barcids" and the Carthaginian government was rather loose and one can see that the Carthaginian empire was perhaps not cohesive enough, whether at that time or before, to undergo large scale, deliberate assimilation.

I hope that gives some satisfactory answers!