For disclosure, I'm a BCG consultant and so my opinion here is subjective.
If you manage to receive an offer from more than one out of McKinsey, BCG and Bain, congratulations! You're in an enviable position. They are all excellent firms with an incredible reputation.
In my opinion, the differences are slight and are likely to vary most with the culture of the particular country rather than firm. If you have to make a choice, my advice is to meet as many people from those firms as possible and make an opinion based on the type of people you have met. You will spend a long time working alongside them, and I've found the biggest predictor of happiness on a project is the people you work with, not the client, topic or industry. it's important that you get along with, and are motivated by your colleagues.
However, here are a few thoughts:
In terms of approach to work, it is essentially the same top-down hypothesis-driven method.
Unlike other industries (such as investment banking), it is very rare for consultants to move between firms. Hence the difficulty in gaining a true comparison beyond the stereotypes presented in forums such as this one. I have a friend (junior) that moved from McKinsey to BCG, and found that McKinsey had a more competitive culture, whereas BCG was more cooperative amongst peers, and also with clients. I have worked with another freelance consultant that works with both McKinsey and BCG who has supported this view. Keep in mind, this is only two data points.
McKinsey, BCG and Bain all work with similar types of clients. However, in different geographies, firms may do more work in one industry than others. But the biggest driver of the type of work is driven more by the economy of the particular country rather than firms. A firm is more likely to do oil and gas projects in Northern Europe, Canada or Brazil than in other locations. This will change with economic and industry cycles. For example, in Australia, mining was significant during a boom in the late 2000s, and declined post-GFC. Some differences do occur across firms - e.g. Bain has a bigger PE practice than others. It's best to speak with people at the firms in your geography to find out what type of work they do most. However, I find that the people you work with are more important than the industry you work in.
One thing to note is that structure affects behaviour. McKinsey has a global P&L, which may lead to lead to greater international mobility. Although, there are plenty of opportunities for international mobility at BCG and Bain (perhaps too many).
Overall, this advice really only applies to people with multiple offers.
If you have multiple offers, speak with as many people at those firms and find the people who you want to work with the most.