What is meant by "more powerful"? Does that simply mean that it can do things that a TM can't?
There are input-output mappings that a TM cannot compute. A good example is a mapping that will read in a Java program and output "yes" if the program will eventually generate output and "no" if it will not. It is provably impossible to write a Java program that can make this determination correctly in all cases. It is also provably impossible to write a TM program that can make this determination. On the other hand, we can invent a magic-pixie based system that can do that.
After all, the heart and the liver can do something a TM can't, as can a hammer, so why is it that we assume the brain cannot?
Because the behavior of a hammer is not typically described in terms of computation or of input/output relationships, while the brain is. If you describe what a hammer does in terms of input-output relationships, then a TM can indeed solve a problem isomorphic to the input-output relationships characteristic of hammers.
Are we talking about only the lowest-level outputs? If so, does it necessarily follow that there can be no loss of functionality for higher-level tasks?
There is no such thing as "lowest-level" or "highest-level" outputs. Input is input, output is output.
For instance, a computer that plays digital movies loses the ability to "play movies" when running very slowly, even though it performs all its calculations.
It does not. In fact, that's how a lot of CGI movies are created. One frame at a time, taking several seconds or minutes per frame. Burn it onto a DVD and watch it, and it looks just as the director intended.