So my question is: what is it that human mind can do which a computer (Universal TM) can not?
Let us presume that you set aside the obvious retort: human brains are embodied and have control mechanisms like the endocrine system which regulate behavior based on emotion. Also, let us set aside the difference in performance characteristics between biological-based computation (think professor with pencil) and a supercomputer. Clearly the rate of computation and the lack of errors a computer makes clearly differentiates it from a professor. As currently constructed most computers are built around a high-speed ALU. Almost the entire architecture of a computer or network is meant to serve the purpose of shuffling around data and instructions to feed the ALU which transforms those data according to a set of instructions delimited by the op codes of the processor; often these days they have multiple cores allowing parallel threads to be executed at frequencies which are staggering incomprehensible compared to human minds.
The difference comes to the act of how a computer solves problems, and how a brain does; and to be forthright, there is no exact understanding of what the limits of a Turing machine are in simulating the performance of the brain, and part of the reason is the brain is not entirely understood. With more than more than 200 neurotransmitters and approximately 86,000,000,000 neurons, the complexity of the brain eclipses most supercomputers in ways hard to describe. Remember, unlike a silicon wafer which is designed and etched, the brain accumulates matter and differentiates itself through proteomic expression beginning with the genes of a single cell: the zygote. While logical circuits like the FPGA have an impressive ability to differentiate themselves, no electronic product on the market remotely resembles the brain, it's growth from a neural tube, and it's plasticity.
PHILOSOPHY AND ARTIFICIAL INTELLIGENCE
One aspect of the human brain is that it does have the capacity to function like a Turing machine, although to be more accurate, the Turing machine is merely a description of one ability of the human brain. Automata theory is a fascinating formalism that captures aspects of the behavior of brains, including its use of language, as evidenced by generative grammars and the models that the Chomskian hierarchy provides. The philosopher Daniel Dennett made a great effort in his Consciousness Explained to make accessible an analysis of the origin and nature of the conscious mind and it's relation to the brain. Dennett attacks the Cartesian duality as an Oxfordian, proposes heterphenomenology, and pushes his views as an eliminative materialist including his "disqualification of qualia".
One outstanding question which is related to Searle's controversial Chinese room (to which whole books have been devoted) is whether or not Turing machines can give rise to minds and intentionality. (I made a brief argument for it on SE here). Searle's position is that Turing machines cannot. Many philosophers of AI rely on the fact that there is nothing inherently special about the nature of matter in systems and that the same properties should arise from systems if they are functionally equivalent. Of course, any progress towards having Turing machines function as human minds has been slow going. Nils Nilsson covers the dream of creating minds from matter in his The Quest for Artificial Intelligence. One of the most famous critics is Hubert Dreyfus's What Computers Still Can't Do: A Critique of Artificial Reason.
STRENGTHS OF THE MIND
It does behoove us to take a look at the current state of technology, and some skills human brains have over their artificial counterparts. One of the most obvious was recognized by Pascal:
We must see the matter at once, at one glance, and not by a process of reasoning, at least to a certain degree... Mathematicians wish to treat matters of perception mathematically, and make themselves ridiculous... the mind... does it tacitly, naturally, and without technical rules. (Pensees)
Human brains take in a tremendous amount of data through transduction of their physical embodiments. The number of cells devoted to sight, sound, taste, smell, touch, proprioception, and so on is staggering from a design perspective. Part of the reason the human brain has so many cells is that vast areas are devoted to encoding experience from sensation, the visual cortex being a terrific example. Perhaps best studied, the visual system dwarfs any camera system manufactured. In principle, parallel Turing machines can be constructed, but until we know how and why to construct them, for the moment, the brain has a tremendous advantage in parallelism. In fact, some have claimed that the complexity goes beyond the ability of the human mind to comprehend and that evolutionary algorithms and artificial neural networks (ANN) are the only technique that might get us there. ANNs for instance have been constructed that are content-accessible instead of address accessible. These constructs have the property of association, much like our brains work, built into their input-output performance.
From that complexity and parallelism, it's not clear how Turing machines can be constructed or evolved to exhibit dispositions that brains excel at. One of the most promising areas of research, however, is that of machine learning which can be divided into strategies such as reinforcement, supervised, and unsupervised learning techniques, many of which mimic how the brain learns in some regard.
So, the philosophical implications of the Church-Turing thesis are still an area of active research and conjecture, and no one can speak with any authority on the limits of the von Neumann architecture to emulate the human brain. Few serious analytical philosophical thinkers reject the notion that the brain is a biological computation device. John Searle of Chinese room fame in his The Mystery of Consciousness even accepts the fact that at least part of the brain functions as a Turing Machine. What is not clear is the exact nature of how the mind arises from the brain with all of it's ontological and epistemological implications. In principle, there is nothing that excludes a powerful computer becoming sentient, but in practice, it certainly hasn't been achieved. The brain has the capacity to create a flexible representation of external reality, use it in to solve problems intuitionally (without self-knowledge), and create and share language to communicate about it ways that computers just can't pass muster. For all of its many flaws, the Turing Test still demonstrates how little comprehension computers have in the simplest of conversations, although projects like IBM Watson continue to chip away at humanity's claim to a monopoly on knowledge-how and knowledge-that.