What's the difference between "Mobile-first Responsive" and "Twitter Bootstrap" on Initializr?

Bootstrap is mobile. Mobile first means that you design for a mobile user first and then adapt as the window resizes. Thus, there is no difference between "mobile-first" and Bootstrap. The difference between Responsive and Bootstrap is because Bootstrap is the foundation, and Responsive is a more or less dumb version - less bells and whistles if you don't need a bunch of elements.