Proto-Sinaitic split into two branches: a northern one which spawned almost all the writing systems of the modern world, and a southern one that did not. Perhaps it is fairer to say that one branch of the script went to Yemen and stayed there unmolested.
South Arabian is the only child in the southern branch. (If you look at a map of Yemen, it is kind of remote. It isn’t that far from the Levant by boat now, but before the Suez Canal, you couldn’t get from the Mediterranean to Yemen without going though a bunch of desert. It is not at all far from Somalia by boat, but you do have to have a boat.) South Arabian happily sat in Yemen and evolved. By about 800 BC, the South Arabian script had developed quite distinct glyph shapes.
Even in the far-off reaches of Yemen, however, South Arabian was able to borrow good concepts from other languages. It adopted the matres lectionis idea, presumably from Aramaic, and the alphabetical ordering from Ugaritic. However, the alphabetical ordering it got from Ugaritic was NOT “abg…” that turned into the “abc” of Latin, but the other Ugaritic ordering, which went “hlħmqw…”
One innovation that they never did get was vowels. South Arabian went to its metaphorical grave without markings for vowels (aside from a few double-duty matres lectionis consonants).
South Arabian could be written left-to-right or right-to-left, with the symbols reversed according to the direction. It also had a cursive form (Zabur) that was used for more casual communications.