The Carthaginians were in several ways distinct from their Phoenician forefathers, while at the same time remaining a recognizably Phoenician offshoot and maintaining cultural ties with Tyre throughout their history. A notable difference is that the Carthaginians were an aristocratic society, while most other Phoenician citiescity states were kingdomshereditary royalties. Furthermore the Carthaginian religion, while a direct continuation of the Phoenician religion, had distinct elements adopted from the local traditions of the civilizations the Carthaginians came into contact with.
I think the more accurate description is that the Carthaginians were a distinct branch of the Phoenician civilization, one that today we commonly refer to as Punics, from their Latin name.
You aren't doing the Phoenicians' much justice though, their civilization is an older one and their high point is within the Greek Dark Ages, an enigmatic era for which we know very little. The Phoenicians, or perhaps more accurately their Philistine ancestors, are a very strong candidate for the Sea Peoples that terrorized Egypt and Greece at the time, and while no record of a general comparable to Hannibal survives, there's little doubt that they were both feared and respected by their contemporaries.