As a native speaker, I may actually not be the best person to help! Maybe someone who has had to study the grammar and understand it as an outsider would know the difference.
I think, if you're using it as an adjective, "northern" is normally better, but you could say "in northern Sweden", "in the north of Sweden" or "in north Sweden" and all would sound natural. If there is a distinction, perhaps "north Sweden" makes me think there might be a formally defined region (say Norrland) rather than a more vague sense of "generally in the northern half". Normally if there is a precise boundary, I would use "North", and the same if there is a "north X" and "south X", while "northern" is more descriptive than precise.
Proper names (North Korea, North Carolina, North Berwick) are normally North, though Northern Ireland is an exception.
Northern is an adjective meaning "the area/part that is at the north of". North is the direction itself. Saying "North Sweden" would sound unnatural and wouldn't actually make sense. While, "North India" would sound as sensible as "Northern India". But, "North Texas" sounds more sensible than "Northern Texas". It chiefly depends on what the people of the area like to be called. As for the continents, "North Asia", "Northern Europe", "North Oceania", "North America" (Or "North of North America" and "North of South America"), and "Northern Africa" sound plausible.