This is not quite the right question. The right question is "How do interviews with partners / principals" differ from earlier round interviews? They differ in a few ways.
- Fit: Partners have been in the firm a long time (many since University or Business School) and consider it a very special place. They put more emphasis on "Why consulting" and especially "Why their firm".
- Experiences / Personal Impact: Partners are looking for candidates that will become future partners and view your past experiences as indicative of future performance in the firm. So, they are looking for examples of past experiences that paint the picture for them of how you are going to perform in the firm and how that is going to make you succeed all the way to partner
- Cases: First, unlike more junior consultants that interviewed you in the 1st/2nd round, Partners usually haven't been through interview training. So, they are less apt to "go by the book" in their scoring. The cases they use don't always come from the active case library and can be based on their current projects or current events. Second, they are trying to transpose your performance to a client situation. For instance, during a quantitative problem, they judge the setup and walkthrough as importantly as coming up with the right answer because they are trying to imagine you walking their long-time client through a quantitative session (e.g., how we got to the -3% cost reduction figure) and assessing whether you would confuse the client or the client would come out confident in the #s.
So, whereas #3 is most important in early rounds, more emphasis is placed on #1 and #2 in final rounds.