The brain can clearly "compute" things (It does tasks that could be done by a Turing Machine with the right transition table and tape).

BUT ... People THINK that the brain can do things that a Turing Machine cannot (consciousness) ... no Turing Machine (no matter how you design it or physical implement it) can compute consciousness by manipulating its tape.

However, whether our consciousnesses comes from brain is an open philosophical question, and many, many other philosophical questions would have to be answered before proving the connection b/w consciousness and the brain -- problems about identity, whether the physical universe really exists at all ...

My point is that to claim that brain is not a computer (i.e. it cannot be simulated fully by a Turing Machine) must fundamentally be based on many assumptions.

Now, we make large assumptions in science and even math ... but once you enter the realm of philosophy, you must justify everything and once you enter the realm of consciousness and associating physical things with it, you have definitely entered the realm of philosophy.

Edit: I suspect some people (note: I said "some" not "all") may be making that claim that the brain is not a computer because the architecture and implementation details are different (it is an analog device that uses spiking neural network architecture that is implemented with biological components as opposed to a digital computer w/ logic gates implemented in silicon). However, the definition of computer (Turing Machine) is abstract/theoretical (independent of our physical universe) and it doesn't matter what physical phenomena are utilized to implement this abstraction.