The brain is to mind as the eye is to sight.
Brain is the hardware and the mind is the software with the totality in action, so the brain with mind is the hardware plus software. Mind is probably not brain in action alone. It may also be a whole-body phenomenon, with inputs from environment as well. So mind is much more than an active brain. The brain is like the hands that make the bread; the mind has the knowledge of how to do it. The brain belongs to the body, the mind to the heart (soul).
Analyze the mind systematically: As with most analytic philosophy, it has different understandings based on different theories. A general rule of thumb though is that the brain is the physical 'stuff' - neurons, the squishy stuff, grey matter etc. However the mind is the 'non-physical' stuff, such as thoughts, feelings, emotions etc. The idea is that you can't 'touch' a thought, and there are no 'blues' around - they are non-physical and are mental objects (features of the mind). Then you may want to somehow explain the mind which you can start with splitting the problem into different minor questions: For instance, what does it contain; how can we know about it; is it necessarily bound to the body or is it in a different 'space'; can we somehow model it, etc. You may get to a categorized listing of its features, cognitive properties that enable a number of complex things, like consciousness, memory, judgment, ability of making complex decisions, and the list goes on...
Two schools of the philosophy of mind: You may feel to get closer a bit to understand the logic behind the brain and mind, and eventually you would ask if the mind could exist somewhere in a non-physical space. There are two major schools of the philosophy of mind: the dualists, who hold the thoughts of the aforementioned thinkers; and the physicalists, who think that there are no other spaces than the physical, so the mind has to be a property that is describable by basic physical processes (or, at least, it emerges from them in a complex way). Assuming this, we could say, consciousness and the mind is nothing but one property of the hugely complex set of physical processes in the brain (or in the nervous system).
The brain is an organ that organizes information and action across all the cells of a living physical body. The 'small mind’ or ego mind is the vehicle of the emotive, intuitive and semantic content of the organization as it arises through languages out of the context of culture. This represents a tipping point in biological evolution towards new complexity and the emergence of a multiply realized 'thinking' self-awareness phenomenon. Explaining the mind as a set of mental objects is nothing more than a necessary level of abstraction in the game of searching for a model of consciousness. WHY can it not be measured or 'touched'? Can it still be a property of the brain, of the physical brain? Can it emerge from the 'simple' rules of physics? Our understanding of the physical world is so narrow that it would be a surprise if we could understand the workings of the brain deeply enough to explain mental phenomena. A model which is needed whether the mental is a separate substance or it is physical. It depends on the nature of 'non-physical'. What we call non-physical today, may turn out to be very physical tomorrow. That would only indicate that today's physics is very limited. On the other hand, though, there is a possibility that mental phenomena is not physical, therefore searching it inside the brain (in a physical sense) would show nothing valuable.
The brain is physical and tangible, but the mind is abstract and philosophical. The philosophy of mind will always be useful to the gaining of knowledge and ultimately capturing the insight and wisdom, and it will undergo many paradigm shifts as time goes on, as with our understanding of 'non-physical' phenomena like mind.