Since I am not sure what you are referring to in your first question (“Why did the Visigoths flee from the Ostrogoths), I will try to give a general idea of what happened on the Danube front in the late 4th century (the period where the interactions you seem to mention unravelled). It should be said, first, that our main sources for the events of the fourth century do not mention Visigoths nor Ostrogoths. Later on, in the 6th century, Jordanes, a Gothic historian from Ostrogothic Italy, composed a text called the Getica, in which he projected the notions of his day (Ostro/Visi) onto the realities of the 4th century. Ostrogoths and Visigoths proper, as I will try to show, did not appear until after the confrontation with the Huns.

Ethnicity is complicated

I have to make another preliminary point about ethnicity. We usually tend to understand “barbarian peoples” as cohesive groups, sharing one ancestry and a well-defined identity, probably inherited since the dawn of time (so that if Ostrogoths existed in the 6th century, we must be able to trace them back to the 4th century). It is all the more tempting to do so since people of the time apparently endorsed this idea, or, at least, promoted it. Jordanes' Getica (that I mentioned a bit earlier) did try to create an uninterrupted Gothic history, starting in Scandinavia centuries earlier, complete with a long dynasty of kings.

If we look at it a little closer, actually, it becomes clear that ethnicity was nowhere as clear, especially in the Migration period. In fact, there was a considerable amount of political will in the way one “created” a people. What is more, the highly unstable context of the period, where political leaders and warlords could reap immense gains in the Roman Empire, partake in the political culture of their prestigious neighbours, but also risked to be crushed by other contenders, means that groupings could be dissolved and recreated; more frequently, existing groupings incorporated many elements that had no direct link to them, other than the political will to regroup in order to form a more powerful entity. All this may sound a little abstract, but the Goths actually are a perfect example of all these dynamics.

The Huns were dangerous people

In the later part of the 4th century, as you may know, a new player began to influence Eastern Europe: the Huns. The Huns came from a part of Eurasia that played a crucial role in its history: the Eurasian steppe. The steppe is a large band of grassland that straddles Asia and Europe, from Ukraine to Mongolia. It is a tough environment, whose inhabitants adopted a mostly nomadic lifestyle. Horses are particularly suited to the steppe; steppe nomads themselves were apt horsemen, a quality that could easily be converted into military power. Steppe society also was very fragmented in normal times, but it could at times be gathered under the banner of a leader (think Genghis Khan). A successful group could then “snowball” and agregate various forces to attack the sedentary world, an attractive source of wealth and loot.

Steppe nomads were not unstoppable, but they were very powerful foes. There are doubts about the power of the Huns; usual interpretations tend to minimise their impact (when compared to other steppe powers); there was a recent attempt to revise that traditional understanding (in The Huns, Rome and the Birth of Europe). It is clear, though, that they were powerful enough to threaten peoples of Eastern Europe. Ammianus Marcellinus, a Roman historian of Greek origin (who was writing soon after the events), writes that : He [Ermanaric, prince of the Goths] was astonished at the violence of this sudden tempest, and although, like a prince whose power was well established he long attempted to hold his ground, he was at last overpowered by a dread of the evils impending over his country, which were exaggerated by common report […]. (it is not entirely clear what Ermanaric's exact status was, and it is not even sure that it is meaningful to pin it down; he clearly was a dominant noble, probably what we might call a king, but it is also probable that his authority could be contested by other important families, and depended on results rather than on a well-established line).

The making of the Ostrogoths/Visigoths

In 376, in light of the impending Hunnic invasion (a Gothic king/prince had already been defeated), some groups of Goths decided to cross the border. Ammianus Marcellinus, at that time, makes a distinction between two groups: Tervingi and Greutungi (which had existed for some time, and were already attested at the time of Diocletian and Maximian, roughly a century earlier; it is plausible that they were geographical names). A group of Tervingi and a group of Greutungi crossed the Danube at the same time, but under different leaders. They later coalesced (under Tervingi leadership) in order to face the Roman forces, that wanted to keep them away from their territory. After having defeated a Roman army (and killed an emperor) in 378, they were settled in Roman territory in 382. However, a Goth named Alaric lead a revolt in 395. There has been some debate, but it seems that most of his troops were levied from unsatisfied Goths that had been settled in 382.

This group was the basis of what was to become the “Visigoths”. It was identified with the Tervingi as early as c. 390 (in the Notitia dignitatum and the Historia Augusta); however, this probably was a shorthand. Former Tervingi certainly were the core of the new “Visigoths” (that is « Wise Goths », according to the most prevalent interpretation), but other Goths and even people with a non-Gothic background were incorporated in what was to become a new ethnicity. In the meantime, Goths that had chosen to remain beyond the Roman border were integrated into the Hunnic army. They would eventually became known as the “Ostrogoths” (and were themselves linked with the Greutungi, though they actually did not represent all the Greutungi). To complicate matters further, it seems that these new identities (Ostro/Visi) were themselves re-interpretated in the 6th century, to fit the new political map (Visigoths in Spain, Ostrogoths in Italy), with a new interpretation (Visi = West, Ostro = East).

I will now try to make this long story short: Ostrogoths and Visigoths are not pertinent concepts during the times of the Hunnic attacks on Eastern Europe in the 370s. In fact, these groups would come into being as consequences of these attacks. Goths were defeated by Huns because steppe warriors were formidable enemies. However, though there certainely were internecine conflicts within all people who self-identified as “Goths”, one cannot say that “Visigoths” (even if we equate Visigoths with Tervingi) fled from the “Ostrogoths”.