There is a joke among linguists about the difference between a dialect and a language: “a language is a dialect with a standing army”.
Similarly, I think that the distinction between the first alphabet used to write Hebrew — what is commonly called “Old Hebrew script” — and Phoenician script is more a description of a cultural difference than a functional difference. To my eye, Old Hebrew script looks more like Phoenician script than some of the different dialects of what is happily called Phoenician script.
The Samaritans, like the other tribes of Israel, started out using the Phoenician/Old Hebrew alphabet. Most of the tribes of Israel eventually switched over to using the Aramaic alphabet, but the Samaritans kept using the Phoenician/Old Hebrew alphabet. Over time, its glyphs evolved into a distinct variant. Note that there is still a one-to-one correspondence between Phoenician/Old Hebrew and Samaritan, so Samaritan has more of a font difference with Phoenician than a fundamental difference in the writing system mechanics.
The alert reader will notice that I used the present tense in the last sentence: there are still about 700 speakers of Samaritan in Israel and the West Bank. The glyph at the top is the modern version of the “yodh” character.