Why is the Phoenician alphabet (modified by the Greeks and then the Romans) still used in most Western cultures today more than any other form of ancient writing?
The major reason for this is simple geographic luck.
The Phoenician alphabet, of course, was part of the Mediterranean world because the Phoenicians were such important traders. It spread across the region to Greece and then to Rome. Rome just happened to be the civilization that ended up with a huge empire in Europe and its alphabet, therefore, became the most used alphabet in the world.
The nature of the Phoenician alphabet may have played a role as well. The fact that it did not contain "logographic" symbols like Egyptian hieroglyphics made it easier to use. However, the main reason that it has come to be used by Western cultures is simply the fact that it was the alphabet used by traders and therefore it came to be used by the Romans. The Romans, in turn, spread it across Europe.