A feature of the post-Roman period in Western Europe and North Africa is the religious difference between the native Trinitarian Christians and the Germanic Arian rulers. Generally this didn't lead to much conflict with the notable exception of North Africa.
I understand the basic theological distinction, but did this lead to any differences in lifestyle, religious practice, or views about what Christianity required of its adherents within these religiously divided societies?
While the Christian credo of most of Vth century Barbarians is commonly called Arianism, it was actually quite distinct from the doctrine preached by Arius and more exactly described as Homoianism which is the term Goths used themselves, calling their credo 𐌲𐌰𐌻𐌴𐌹𐌺𐍃 (galeiks), a direct translation of όμοιος (homouious) meaning "similar", i.e. Father and Son being similar in essence but not identical. Also called Subordinationism, understanding the Son as subordinate to the Father, it was a form of compromise theology between Arian teaching and Nicean credo. Rather than preaching that the Father and only Him was divine, Homoianism underscored a shared divinity but an unequal standing for the Father, the Son and the Holy Spirit, the first being the dominant and first member of the trinity.
It was originally developed in the imperial court and entourage, mostly to resolve the differences after the council of Nicea failed to do so entierely, and this is when it got labelled as "Arianism" by Nicean followers, while Greutingi Goths converted to the then imperial-backed doctrine as a display of symbolic obedience to the emperor.
When after the Battle of Adrianople, the imperial court favored a strict Nicean doctrine; Homoism was essentially kept by Goths as an identity marker as they further adopted the cultural and social codes of the Romans (considering Barbarians as "Germanics" is already difficult enough in the IVth century, and becomes untenable afterwards) and due to their prestige and an active proselytism to other Barbarians, were eventually imited by most of the other Barbarian peoples or groups in the late IVth and Vth centuries (including peoples as Burgundians or Suevi that converted to Nicean Christianity, but at the notable exception of Franks). It is possible that Homoianism remained a thing among Romans for a time but its association with Barbarians and defeated emperors made a quick number out of it on this regard : religious tensions were limited, if existing at all, essentially because it was such a social and political marker, more than cultural or religious. It was not that post-Imperial societies were much religiously divided than they were socially and politically divided.
The Vandals do represent an important exception, but one that should be put into a general context : contrary to what probably happened in the rest of western Romania, where Barbarians took over part of landed fiscal production (rather than part of the land directly), Vandals likely requisitioned lands of Africa Proconsularis, expelling Roman landowners (a thorn in the reputation of Vandals until the end) and likewise, attempted to break the power of Nicean bishops. Even there, the persecution doesn't seem to have been either as fanatical than Victor of Vita made it, and it wasn't systematical or really long-lasting.
Homoianism was eventually not really distinct from Niceanism, especially as followers of its credo discouraged theological refinements as God was essentially mysterious (in the religious sense) and not understandable by human intellect. While the political advantage not dwelling too much into christology and giving ground to Orthodox criticism as rejection of eastern "innovations" was a good part of western religious identity; it might be coming too from a greater biblical litteralism from Homoians than Niceans.
This theological proximity and prudence made the few details about Homoian liturgy rather unimpressive and with limited differences : people sang different hymns, quite possibly in Gothic (at least for Goths) as while it felt in disuse as soon as the late Vth century or early VIth century it kept ceremonial uses; maybe a different number and use of chalices during services but we don't know much more about it.
It's possible that this doctrine held a different view on sainthood as well : being careful to distinguish them from divinity, possibly frowning upon doulia (veneration of saints and their relics), they celebrated same saints as Niceans but at different dates, which is possibly due to a different liturgical calendar, and had furthermore their own saints. This difference might be the reason why Nicean bishops and abbots believed that their own stance on sainthood and their veneration was a useful counter-point to Homoians.
Overall, while differences in a religious service would have been perceptible, they seem to have remained relatively superficial : a conversion from an Homoian credo to Nicean's simply required a professio and not a baptism (while it did happened sometimes, it was more or less frowned upon by the Church). Public religious debates still happened from time to time, but more for entertainment and old habits' sake : religion was the matter of the Church and State, and tensions really appeared when politics where involved. Homoians were more or less supposed to adopt the correct form of Christianity eventually, as their mistake was much less important for the Church than heterodox beliefs among Romans (such as Pelasgianism or semi-Pelasgianism) which set a bad example : such view wasn't necessarily illusory in the VIth century, and it seems that Alaric II was on the verge of converting to a Nicean credo or at least, promote a further compromise (which would have led to a "Niceanisation" of Barbarians anyway) before the Battle of Vouillé.
Homoian Barbarians kings protected a form of religious diversity out of political identity (up to professing a "you have your truth, I have my truth" attitude) until the VIIth century where association between the State and the Church became too tight and as the distinction between Barbarian and Roman lost political relevance after decades of social fusion; and where notions of political and religious unity dominated.
- Arianism, Roman heresy and Barbarian creed, edited by Guido M. Brendt and Roland Steinarcher
- Les Racines Chrétiennes de l'Europe, conversion et liberté dans les Royaumes Barbares, Vème -VIIème siècle, Bruno Dumézil