*Please note that this response is a generalization and differs across practices, geographies, and individuals. I also am not speaking on behalf of either organization and am presenting a viewpoint from materials I have read and my experiences.
Looking to get specific on your answer, it's because of the company cultures/values and ways of working. Cases represent what you'll do on the job, and so, the styles are adjusted to how you'll operate in each company.
Ways of Working:
BCG: Belief in taking the client through the journey. They believe in solutioning with the client to come to a conclusion/solution/implementation together. This means "bespoke" solutions that are tailored specifically to the client. They believe that, while this generally takes longer and has more roadblocks, it leads to real buy-in and committment to to end-product, which allows for a stronger result and one that lasts after BCG leaves. The negative view is that their answer is "what the client wants".
McKinsey: Belief in applying tried and true methods that they know work. They believe that solutions are better arrived at without undue influence from the bias that may come from close interaction with clients. They believe that better solutions can be achieved faster through this. The negative view is that their approach is "dust off another report"
The Result for Cases:
BCG: Creative, "out of the box" thinking desired. Candidate-led style as you'll need to drive the client and adjust to their responses AND "flatter" company structure means you need to drive your own module.
McKinsey: You better know your frameworks. Apply grounded principles to a case and be far more structured. Interview-led as more senior people as company will tell you what they want done.
I could go on, but I think you get the point :)