Hi there! First off huge congrats on getting both offers. This is not super common. The below is my two cents (without much bias, I was at McKinsey).

The short the answer is you should go with whichever firm's people you liked/enjoyed meeting the most. As such I would suggest asking to meet as many folks at both places as possible (potentially in practices you're excited about) across different levels. At the end of the day this is a people business and you will be spending most of your time with colleagues in a team room/on the road so it's important you get on with them.

Some key differences to bear in mind (that I have heard anecdotally and having interacted with friends at all levels at all three firms), this will of course differ by office. 

1) Industry expertise: Whilst both consulting firms are generalist full service firms:

• Typically Bain will win and serve on the highest profile private equity/principal investor projects (even vs McKinsey)
• Bain typically is not known for public sector work, whereas BCG does typically does a lot
• BCG (via BCG DV) will do a lot of digital work & digital business building
• On other kind of projects (e.g., classic private sector strategy pieces/transformations) net net globally, you will get on par exposure between the two firms

2) Exits: Having just been through a buyside exit process, PE/growth funds, typically prefer Bain, given the PE focus; so if you think PE/the buyside might be for you after consulting, this is something to consider (this is not saying you can't do it from BCG, perhaps on the margin easier from Bain). Other exits, I would say are equally accessible from both firms.

3) Staffing: 

• My understanding, is that Bain staffing model (at least for most juniors) is more ‘command and control’ (at least earlier on) vs. BCG, where you can have slightly more control over the type of projects you do (via networking etc. vs. Bain where you express preferences to your staffer as opposed to actively finding projects). This is therefore personal preference depending on how much you enjoy networking/have strong preferences on what you want to do. 
• Typically, Bain staffs more locally than BCG, where it's more regional, so you maybe travelling a little more with BCG