In many cases, there's no difference at all. Where there is one, it's generally that the term "brain" refers to the physical object inside your head while the term "mind" refers to the human capacity to think, sense, and process information.
So you wouldn't say, other than if you were joking or trying to make a philosophical point, "My mind weighs about 6 pounds." Similarly, you couldn't say, "I believe the human brain is actually a supernatural object where thought occurs and should not be considered part of one's physical body."
The substantial equivalence of these two terms flows from the well-known, biological fact that the human brain is the organ responsible for our ability to think, sense, and process information. For people who don't believe that, the terms become more different.