Why are Turing Machines important?
It has been mentioned that Turing Machines only emulate one aspect of a computer. So why are they so important to the field of computer science?
The answer is that Turing Machines are the simplest construct that is able to implement algorithms. Obviously algorithms can include certain things which we don't think of as being able to be written on a tape solved symbolically. Yet, if there exists a problem of any type that is difficult to solve, and if that problem can be solved by an algorithm (ie. a sequence of finite steps), then a Turing Machine can represent the difficult part.
So how does a Turing Machine relate to a computer?
Well a computer is hardware (ie. circuitry) that provides a physical model that can emulate a Turing Machine. In reality we like computers that have things like monitors, a memory hierarchy, networks, keyboards, and all manner of other peripherals. The problem of building hardware that does any number of useful physical tasks is what the field of Electrical Engineering is all about. But for all the electonics we've created, none of it can solve a single theoretical problem that a Turing Machine can't solve.
The practical significance is that all programming languages are capable of solving the same set of a problems. A Turing Machine is simply easier to theorize about because it has the simplest possible definition. By proving that a problem is Turing Decidable, Turing Recognizable or neither, we prove things about the limitations of computers. Proving things directly about a computer or programming language would be infinitely harder and not any more useful.
What about unusual electronic constructs like neural nets?
To the best of my knowledge, Neural Networks are still built entirely out of hardware that is predictable. Anything that's behaviour can be predicted (even if only on a theoretical level) can be represented as a Turing Machine.
To say that the Turing Machine represents the ultimately complete model of computation may prove to be presumptious, however, as human knowledge has never been as complete as it seems at the time. There may be some non-algorithmic method of computation may be possible which render greater power than a Turing Machine. I realize that probably sounds ridiculously outlandish to any computer scientists, but I'm just throwing it out there since questioning the most basic assumptions is how the greatest progress is usually made.