Well, they were rivals and often enemies.

Dont forget aeneas supposedly sheltered with Dido in Carthage after the fall of Troy. ie even in such ancient times Phoenicians were allied to the enemies of Greece. Then of course the Persians used the Phoenician navy quite extensively during their empire. Again these often fought directly against the Greeks, espeically the Athenians. If I remember right various Athenian excursions to Egypt (supporting revolts against the Persians) were defeated by navies containing Phoenician ships. And dont forget Alexander's very bloody conquest of Tyre and Sidon.

Greeks set a lot of store by this kind of ancient history. Im sure the Phoenicians did likewise...

Also, the commerical rivalry cant be underestimated. In many parts of the world it would have been a familiar sight for greek and phoenician traders competing for the same goods.