from my experience in doing recruiting for both McKinsey and BCG, and having interviewed many applicants in both styles, I believe I can give a comment here. :)
The difference is essentially rooted in two different philosophies.
Candidate-led case style (BCG et. al): Here, the candidate's conceptual thinking ability, his ability to hypothesize and prioritize, as well as his leadership capacity to drive problem solving forward are under scrutiny. The interviewer will try to get a holistic picture of the candidate! And over the course of 4-6 interviews, consistency of the candidate's process rigour is verified. Most firms have adopted this style.
Interviewer-led case style (McKinsey): McKinsey believes that a defined and limited set of qualities and abilities determine a consultant's success within the Firm. Hence, interviewers double down on these abilities in order to test them in a very focused way. This leads to the interviewer-led case style, which sometimes feels like multiple mini-cases under a common thematic umbrella. You can also see this philosphy in how McKinsey runs the fit part, which is much more focused compared to other firms: the Personal Experience Interview (PEI) at McK scrutinizes 3 defined dimensions in extreme detail (vs. more open discussion formats at other firms like BCG).
That being said, the difference between the interviewing styles of both firms is not that clear! Also at McKinsey, you might encounter candidate-led cases - especially in later rounds with more senior interviewers who often do not adhere to a "script" but prefer to open the conversation after the basic problem solving boxes have been "ticked" in previous rounds.