To clarify, the territory of Bradenburg was part of the HRE, but Prussia was not. Frederick William (Grandfather of Frederick William I), the Great Elector, helped to unify these territories. How did this situation play out politically and economically?
For example, did Brandenburg have to take certain orders from the Holy Roman Emperor, when Prussia did not? Furthermore, if so, how could Brandenburg-Prussia be unified when the territories are under the authority of different people?
The head of the House of Hohenzollern served a dual role as the elector of Brandenburg, and duke (later king in) Prussia for several decades. Essentially, this meant that the Hohenzollern rulers enjoyed more sovereign powers in their Prussian holds than in their Brandenburg ones. This increased control could be attributed to the earlier decline in central authority of the Polish-Lithuanian Empire than the HRE. Brandenburg was not one of the traditional Dukedoms of the HRE, and the Holy Roman Emperor only made concessions to the rulers of Brandenburg as it become less practical to exercise direct control over them.
This is also the reason why the Hohenzollerns originally become Kings in Prussia rather than Kings of Prussia. Within the borders of Prussia, they could reign as Kings, but in their Brandenburg holdings they only exercised the traditional powers of an elector, and were subordinate to the Holy Roman Emperor. It was not until Fredrick the Great's victories in the War of Austrian Succession that the Hohenzollerns exercised enough autonomy that Fredrick could declare himself the King of Prussia outright.