I have been working in the BCG office and I did the interviews also for Bain and Mckinsey, where I have many friends. It's really hard to summary the differences between these 3 offices but I will tray to address your points:
- Culture: The working culture is really similar in each office, the customer comes first, no matter what. Mckinsey is maybe still the office where you feel "more exclusive" but probably also the more though for work. Bain and BCG have developed a little "start up" attitude with some small initiatives that make your life better.
- Growing: BCG has been impressively growing in the past years (double digit per year), while McKinsey has slowed down a bit in Italy . Bain has just joined the international network and is in the middle as growth.
- Type of projects: I would say no difference between McK and BCG (there are often teams of both companies for the same client). Bain is doing these kind of projects but also differentiating a bit focusing on small projects (that's why you could experience double staffing even as new hire)
All of them are excellent offices and place to work, but you should try to understand which office is closer to your personality, also looking around and knowing people working there during the interviews. One aspect that you should consider is also the MBA policies, that are quite different between them.
I know that is difficult to understand differences between them from outside, I had really hard time to decide which offer to accept.
Feel free to contact me if you want to discuss more about it, I can share with you which were the drives of my decision and some general insights.