Congratulations on getting past the 1st interview round! The key differences between the 1st and 2nd interview rounds, besides the interviewers being principals or partners as you mentioned, are in structure, focus and selectivity. Context wise, I'm speaking from my experiences at McK and BCG, where I passed both interview rounds en route to getting offers from both firms.
The structure in the 2nd round will most likely not follow the standard fit + case interview format. Partners may prefer to launch into a conversation about the latest developments in a particular business, industry or geography and talk about your motivations and life experiences later. Note they'll still expect you to layout your thoughts and communicate "top down" by leading with headline "insight" statements. This is the difference between being structured yet adaptable and being formulaic.
The focus in the 2nd round will likely be influenced by your performance in the previous round. The 2nd round interviewers will test on any areas where they perceived you needed to work on (e.g., more quantitative problem solving if you struggled with math in the 1st round, more creative brainstorming if you had difficulty coming up with tailored approaches and ideas to problems in the 1st round, etc.).
The selectivity in the 2nd round is much lower, as this is the round where hiring decisions are made. It may seem obvious but the stakes are higher and it requires all 3 partners to agree that you've met the bar, which explains the low acceptance rate (~1-3%) at the top 3 firms.
In short, the 2nd (final) round is more creative, yet structured, more focused on your development, yet still supportive (interviewers are there to help you succeed, not fail) and more selective, yet fair.
My recommendation is to practice unorthodox cases with a focus on your 1st round feedback and on preserving structure & communication no matter the subject or interview style, so that the interviews may catch you unawares but never unprepared.
Best of luck with your interview!