The border between Norway and Sweden (including Finland at the time) was formally established by the Treaty of Strömstad in 1751. However that treaty more or less followed the traditional border (except in Bohuslän, which had been ceded to Sweden in 1658) which ran along the drainage divide. The border had not been very well defined in Lapland as neither government particularly cared about inland Lapland, which had no permanent, farming population, just a sparse amount of semi-nomadic Sámi.

Finland's borders to Sweden were defined when Sweden ceded those areas to Russia in the Fredrikshamn treaty of 1809 and the Muonio and Torne rivers defined a border, where no border had ever existed before. Finland was where Finns lived, Lapland was where Laps (Sámi) lived, and there was single Lapland province, which in turn was only part of the Sámi homelands (Sápmi) which extends into Norway and Russia but was never an independent state or political entity.

So there's no historic cultural difference between Karesuando on the Swedish side of the river and Kaaresuvanto on the Finnish side; They're just Swedish vs Finnish transliterations of the North Sámi name Gárasavvon.

Historically the main language spoken in the area besides Sámi was Meänkieli Finnish. So if you visit for instance Svappavaara in Sweden you'll see a trilingual sign with the town name, none of which are really Swedish. ('Svappavaara' being a Finnish corruption of an older Sámi name, then loaned into Swedish, 'Veaikevárri' the modern North Sami name and 'Vaskivuori' being Meänkieli)