I think given I've done a stint in management consulting now, I'd probably add that a BA role is typically a little more structured around the analysis aspect and typically you have already been given some framework as to what you want to analyse/what decision you are trying to support. There is potential for a lot of depth here too as you are very focused on specific needs.
MC on the other hand is far broader and you haven't got the luxury of guidance or framework and you're very often just coming up with it as you go along (not totally winging it though). You are there to help define the frameworks, the
target operating model, the
end state/goal, etc... and you are able to do that because you've done the analysis/had experience elsewhere. Traditionally the role was more about coming up with the vision but nowadays it's very much about
implementation and execution too, and some skin in the game