1) Scope of work is similar but McKinsey has a few dedicated functional practices which I don't believe BCG has. For instance, McKinsey has an Operations Practice, where the people in this group are focused on primarily Operations projects (supply chain, manufacturing, etc.). Such practices usually include regular training events and have their own requirements (for instance, McKinsey Ops is more focuseded on you contributing to Knowledge creation than Generalist Practice). Not saying BCG doesen't do Ops work, but don't think they have a dedicated practice
2) Difficult to comment on BCG culture since I have not worked there, and I think it's not fair to describe culture without experiencing it. McKinsey really believes in 'Make your own McKinsey'- they provide a lot of flexibility across all levers - function, industry, people, geography and there is a dedicated Professional Development team that helps you get on projects that will be the right fit for you. There is a strong culture of regular and strengths based feedback - both upward and downward. There is an obligation to dissent - sense of hierarchy is low. You're supposed to speak up if you do not agree with something. Obviously, projects can be intense and keep you busy, but if you're vocal about priorities that matter to you (e.g., workout everyday, start and end early, fly out early on Thursday, etc) teams usually make it happen.
3) Benefits are very similar for an entry level role
Hope this helps!