I’m an alum of IIT Patna. So I may not be the right person for an A2A. But instead of declining, I’d give you a little info.
As far as my opinion is concerned, IITP Mech has no flaws in it. Faculty, Curriculum, Projects, Funding and Opportunities, all are good. So is the same with IITBBS Mech (I visited it, to meet one of my friend. So I saw their campus). I am quiet sure that it is similar for electrical guys too. To be honest, both IITP and IITBBS are the same in almost every aspect I can imagine of.
So when it comes to the question of chosing one over the other, your personal preference of the location should matter. (Even Patna and Bbs are the same in most of the aspects!!)
So pick the one whose campus is close to your home or your girlfriend’s place.
But since you are considering Mechanical Dept, it is customary that you do not have a girlfriend. This should clear your confusion.