Greek legend says that a Phoenician, Cadmus, brought writing to the Greeks. This is not hard to believe, as the earliest Greek glyphs look very similar to Phoenician. However, the Greek alphabet had something from the beginning that no other writing system had: full independent vowels. Having a full set glyphs for vowels and a full set of glyphs for consonants made Greek an alphabet instead of abjad or syllabary. All other alphabets derive from the Greek; there are no cases of people independently developing an alphabet (with the possible exception of Elder Futhark or Ogham, though I think that is unlikely).
There have been some syllabaries since that have vowels and syllables (e.g. Japanese), and some writing systems that decorate the consonant to indicate the vowel, but none where consonants and vowels are all there and independent units, except for those that derive from Greek.
It’s not that everyone before the Greeks was stupid: alphabetic writing systems can require many more glyphs to say the same thing than a logographic language, syllabary, or abjad. This is not a virtue when writing technology is either expensive (like papyrus) or awkward (like clay tablets).
Partly, Greeks established complete vowels because they had no choice: the language would be unintelligible without them. In Semitic languages, the vowels are not hugely important. Two different words with the same consonants are almost always related, like “swam” and “swum”: both have to do with splashing around in the water. In Indo-European languages like Greek, vowels are really important. “Drown” and “drawn” refer to radically different concepts!
Originally, Greek was written right-to-left (like Phoenician), then boustrophedon, but after a while they settled on left-to-right. Because Greece had such significant linguistic separation (due to being a bunch of islands), there were significant dialectical differences in both written and spoken Greek. They didn’t settle on a standard alphabet until around 400 BC, when Athens standardized on the Ionian script.
While the Phoenicians spread their writing system along the southern part of the Mediterranean, the Greeks spread it along the northern part, most importantly to Italy.