The Samaritan version is that the Samaritans are the original Israelites (the “northern tribes”, although it was really the Judeans who usurped the name “Israel” from their northern kin) who have worshipped Yahweh since ancient times. This is probably close to the truth. Additionally, they believe the split happened when Eli left Shechem and established a new sanctuary at Shiloh.
The Jewish version is represented by the anti-Samaritan polemics you find in 2 Kings, that Assyria deported the northern Israelites and replaced them with foreign colonists who mixed the true Israelite religion with paganism. (However, we find somewhat differing reports in the Bible, and even the MT doesn’t quite agree with the LXX on what happened.)
Ezra reports opposition by the Samaritans to building of the Jerusalem temple, but the historicity of this is doubtful. Furthermore, the Samaritan leader Sanballat who appears in Nehemiah is placed in the time of Alexander the Great by Josephus, and both stories cannot be true. At any rate, it appears that both Jerusalem and Mt. Gerizim had Yahwist temples in the Persian period (as did Elephantine in Egypt).
Since both groups use the Torah (with some minor differences here and there), it is possible that the Torah was produced as a collaborative work between the priests of both locations. The Samaritan Pentateuch emphasizes the importance of Mt. Gerizim more than the Jewish version, and if I’m not mistaken, the Dead Sea Scrolls support the antiquity of the Samaritan version.
In the Hellenistic period, Jerusalem grew in importance and political power at the expense of Samaria. The definitive split between the two religions may have come in 128 BCE, when John Hyrcanus, the Judean governor and high priest, destroyed Shechem and the temple at Gerizim. There was bitter enmity between the two groups from that time onward.
In addition to the Samaritan Pentateuch, the Samaritans have their own Targum, some theological literature written around the same time as the Talmud, the Samaritan book of Joshua (different from the Jewish one), and the Samaritan Chronicles (different from the Jewish ones).
Passages like the following seem to “predict” that Israel will fall into rebellion and be exiled:
Although once you were as numerous as the stars in heaven, you shall be left few in number, because you did not obey the Lord your God. And just as the Lord took delight in making you prosperous and numerous, so the Lord will take delight in bringing you to ruin and destruction; you shall be plucked off the land that you are entering to possess. The Lord will scatter you among all peoples, from one end of the earth to the other; and there you shall serve other gods, of wood and stone, which neither you nor your ancestors have known. (Deut 28:62-64)
Take this book of the law and put it beside the ark of the covenant of Yahweh your God; let it remain there as a witness against you. For I know well how rebellious and stubborn you are. If you already have been so rebellious toward the Lord while I am still alive among you, how much more after my death! (Deut 31:25–27)
Knauf also thinks the content of the laws in Deuteronomy points to its authorship during the early Persian period, and he proposes that it was written by scribal circles in Mizpah/Bethel and Jerusalem. (“Observations on Judah’s Social and Economic History and the Dating of the Laws in Deuteronomy”, JHS 9/18, 2009)
https://www.youtube.com/watch?v=tRrFrx8-wEg
In “A History of Judaism,” Martin Goodman gives about a one-page summary on Samaritans.
“In practice, Samaritans were were treated by Jews as a separate, and often hostile ethnic group. Jews did not become Samaritans, nor did Samaritans ever become Jews.” Goodman also points to two inscriptions on the Greek island of Delos from the second century BCE, where Samaritans are called “Israelites who send the temple tax to Mount Gerezim.”
The origins of the Samaritans is unclear. “According to Samaritan tradition, down to modern times, the Samaritans are direct descendants of Israel who, having survived the destruction of the northern kingdom of Israel by the Assyrians in the eighth century BCE, and avoided deportation, preserved the Torah of Moses not least by worshipping in the divinely ordained sanctuary of Shechem next to Mount Gerezim.”
The Bible says the Samaritans are descended from non-Israelites who were relocated to Samaria, among whom were Cutheans (colonists from Cuthah). They learned the law of the land and the worship of YHWH from an Israelite priest. John Hyrcanus destroyed their sanctuary in the late second century BCE.
Josephus attributes some fickleness to the Samaritans, that when things were going well for Jews, they were part of the tribe. But when Jews were being persecuted, “they are said to have claimed originally both to have come from Sidon in Phoenicia and to have descended from the Medes and Persians.”
Samaritans were a distinct, but related group. The Torah they shared with the Jews of Yehud (with modifications) would seem to indicate that they had fairly close relations with the returning Judahites during the post-exilic period. I suppose that non-Samaritans also lived in Samaria, without sharing membership in the religious group. Maybe they were called Samarians?