For example, after the death of Karloman, Karl inherited Middle Francia and it directly joined his empire, but Brandenburg was in personal union with Prussia after the Prussian duke died. What is the difference that causes a realm to be integrated upon foreign leadership versus a realm keeping its own laws?
I think you're making a fairly big distinction where in reality there existed very little. Lands came into personal union when the monarch of one inherited the title to those other lands. There could be no personal union without inheritance. There are of course dozens of examples of this: Charles V, James I & IV, Edward III's claims to France etc. etc.
Your real question seems to be that why were lands sometimes integrated into a single kingdom or why they remained legally distinct under a single ruler. The answer to that is that it was almost always the latter case in medieval and early modern Europe - if the King of France was next in line to the throne of Sweden, he would become the king of France and the king of Sweden, not king of United Franco-Sweden. In fact I'm struggling to think of a case during the period where this didn't happen. The idea of integrating previously autonomous territories seems to have been pretty foreign to pre-modern rulers. If anyone can cite some counter-examples, I'd love to hear about them.
As for the one example you do cite in your OP, I'm not sure that really counts. I'm going to assume that you're German, and so that the people you're talking about are those brothers better known in English as Carloman I (768-771) and Charlemagne (768-814) (Karl der Grosse in German). (If I've got that wrong and you're actually referring to another Karl and Karloman, please let me know.) This really isn't a parallel situation to the others mentioned above. This is because Charlemagne and Carloman were operating within the Merovingian framework of partible inheritance, which extended to the kingship. Their father, Pippin the Short, had ruled the unitary Frankish kingdom after re-uniting the fractured Merovingian kingdom through conquest and then crowning himself king, with the support of the Pope, in 750. On Pippin's death in 768, both of his sons were made kings, each with a distinct sphere of influence (roughly, Charlemagne got the coastal provinces and Carloman got the interior). So when Carloman died three years later in 771, it wasn't so much two seperate kingdoms being integrated, but simply one king resuming control over the whole kingdom, rather than sharing the rule with his brother. The real indication that Charlemagne didn't really believe in forcibly integrating autonomous polities into his kingdom comes in 774, when he conquered the Lombard Kingdom of Italy - rather than turning it into a part of the Frankish kingdom, Charlemagne now had two titles: King of the Franks and King of the Lombards. Just like the other personal unions.