This is a really surprisingly complex subject. In any case, here's why your brain is NOT a computer in the way we currently define computers (and thus, why CTM is wrong):
Computers operate purely syntactically...they have no content as far as they're concerned...they only care about symbols and patterns of symbols. That's kind of confusing, no? Here's a famous example, called the Chinese Room.
A philosopher is locked in the Chinese Room and is given a huge packet of instructions and a book with Chinese characters in it. The instructions tell the philosopher to change this symbol to this letter when followed by this, etc--extremely complex instructions, no doubt, but doable. When the philosopher has completed moving the symbols around, as according to the instructions (presumably years later...) he slides it under the door.
As it turns out, the characters that he was sent in with were a question, and our philosopher has written a valid answer. The people on the other side of the Chinese Room exclaim that he must know Chinese!
But in truth, we know that our philosopher has no idea what the question nor answer was, but merely had the correct instructions to manipulate the data appropriately.
Computers are like this. They manipulate symbols, but have no content.
As a result, we should conclude that CTM (computational theory of mind) is false.
But there is an alternative idea gaining popularity, called connectionism....there are a different class of computer like things, called connectionist machines, that work is extremely different ways. I won't go into the details because (A) I honestly am not an expert on them and (B) it would take forever and a half, but fundamentally they are different in that they calibrate themselves (and are calibrated) to correct signals coming in, which of course, brings about a different output....that's a fancy way of saying they learn. Computers can save an image of a face easily and instantly. Human beings and connectionist machines cannot. We can however be much better prepared to deal with new situations.
I should say, however, that if we buy that the personality is a totally physical thing, and there's huge evidence to support that, then we are necessarily committed to the idea that it can be replicated (not that it WILL be replicated, but that it hypothetically could be). There are no physical things that can necessarily never be replicated (otherwise, they wouldn't have existed the first time).