This post from three years ago has the info.

The short answer is that the direction seems to have been influenced by the environment of the writing. Inscribing on stone lent itself to a right-left orientation (for the typical right-handed stone-cutter), while, eventually, writing with ink on papyrus encouraged a left-to-right direction (and this transition happened between the 6th and 5th centuries BCE in Greece, so that by the Hellenistic period, almost all Greek writing was left to right, both on papyrus and on stone).

Edit: I would suggest that painting on the ceramic medium might have also played a roll. Some of the earliest Greek writing we have is from painted pottery, including the famous Cup of Nestor. There is usually a lot of talk of "smudging" ink when writing right to left for a right-hander, but I think that idea belongs more to pottery painting than papyrus. The inscription on Nestor's Cup is boustrophedon, as are many Archaic inscriptions.

Ok, here we go.

As your question kind of tacitly implies, the alphabet was adopted by the Greeks from the Phoenicians/Semites. This has been known for quite some time. So--the heart of your question is "why are there divergent alphabetic traditions?"

Writing direction, stance of letters, etc. were not initially settled in West Semitic writing really until the Iron Age. The early alphabetic script (proto-Canaanite, really) could be written sinistrograde, dextrograde, boustrophedon (i.e., alternating --> and <-- every other line), or in columns. Direction of writing was not settled until the rise of the Phoenician national script during the Iron I period, as exemplified by the Royal Byblian Phoenician inscriptions (the Aḥirom Sarcophagus, Yeḥimilk inscription, etc.). This is the script that was ferried over the Greece and adopted there and then became the Greek and Latin alphabets.

Quoting Kyle McCarter's Johns Hopkins dissertation (The Antiquity of the Greek Alphabet, pp. 103-04):

The first Greek users of the alphabet showed a calm disregard for the inveterate Phoenician tradition of sinistrograde writing. The corpus of early Greek inscriptions includes examples written not only from right to left in their Phoenician fashion, but also from left to right and even boustrophedon...One is tempted to dismiss these peculiarities of stance and direction as Greek innovation if not mismanagement. There is certainly no anticipation of them in the Phoenician inscriptions of the ninth and eighth centuries. But do they correspond to scribal custom at any other point in the history of Phoenician writing? Of course, they do. It is an intriguing fact that the old Canaanite graffiti of the thirteenth and twelfth centuries feature the same ambiguity in direction and sporadic uncertainty of stance which characterize the early Greek inscriptions.

And more from McCarter:

As far back as they can be traced, the Greek scripts exhibit the local or epichoric variety by which an Attic inscription may be distinguished from a Corinthian , a Cretan from a Rhodian, and so on. Still there can be little doubt that all were descended from a single "proto-Greek" alphabet. This is clear from certain features which all the epichoric Greek scripts share but which are absent from the parent Phoenician. Of these, the use of the letters corresponding to five of the Semitic consonants ( ʾ, h, w, y, ʿ ) to represent vowel sounds in Greek (a, e, u, i, o) is only the most obvious.

The Greeks already had their own writing traditions when they adopted the Phoenician script. Of course, too, when they needed letters for sounds not in Semitic, they added their own graphemes to the END of the alphabet (e.g., phi, chi, psi).

Latin and Cyrillic are derivative of Greek and likely just adopted the Greek's dextrograde writing style.