There is a difference between brain and mind. The mind is a subtle machine provided along with the body to the soul.  It is such a machine that continuously generates thoughts, feelings, ideas, perceptions, and stores knowledge and memories.

The brain is the hardware that the mind uses for its functioning.  Someone’s brain may get damaged, but the mind may still continue functioning normally.  This shows that the mind is distinct from the brain.  Again, in the case of plants, they have no brain, and yet they have a mind.  Experiments have shown how they respond to the moods of the gardener and speed up or slow down their growth in response.  This illustrates that even plants possess a mind.

The brain is made from the gross elements – earth, water, fire, air, space.  On the other hand, the mind is subtler than these gross elements.  

The Bhagavad Geeta chapter 7 verse 4, states:

भूमिरापोऽनलो वायु: खं मनो बुद्धिरेव च |
अहङ्कार इतीयं मे भिन्ना प्रकृतिरष्टधा ||

bhūmir-āpo ’nalo vāyuḥ khaṁ mano buddhir eva cha
ahankāra itīyaṁ me bhinnā prakṛitir aṣhṭadhā

Read: This verse commentary by Swami Mukundananda in detail

Here, Shree Krishna has enumerated the mind as separate from the five gross elements.  Modern science has some idea of the nature of the gross brain, but is yet to comprehend the functioning of the subtle mind.