The main area I've looked at where there is a huge difference between computer and human processing is pattern recognition. The human brain has an incredible ability to see patterns from relatively discrete elements. This is central to language and image processing and huge parts of human problem solving. One example is how we either read or listen. We do not see each individual letter (or phonetic element) and work out what word they mean. We spot the whole pattern. A good typist does this process in reverse -- they don't think about each letter when typing a word, they have the entire word (or phrase) as a single element in their cognitive processed. With unfamiliar words, we revert to a more piecemeal approach.
There's all sorts of complicated implications of this. We skip huge amounts of detailed data and make rather large assumptions from the context -- which makes us susceptiable error. But it also makes the processing much faster (to read the word 'faster', for example, I process one thing instead of six letter or two phonetic elements).
One approach that offers some real potential is quantum computing.
Here's an introduction that covers it far better than I can.