I think the comparison of computers and the human brain can be useful. However when it comes to atheism, I believe it is an unfortunate analogy. In fact, it kind of makes the case for religion and life after death.
When a computer dies, it is far from over. Computers are resurrected all the time. Ask any computer repairman. Also, even if the computer is not repaired, all of the information that makes that computer unique (Its soul) can be stored remotely, like in the cloud (computer heaven).
It can be proven logically that living things have a soul, if one takes the meaning of a soul to be that which survives an organism after its death. Certainly the memories that a person has left survive their death. So there is at least a soul as long as there are others to remember them. In Billy Joel’s lullaby, he says “lullabies go on and on. they never die that’s how you and I shall be”. So the works of a person survive their death also. In the case of a great artist or scientist, these works can have a tremendous influence on millions. Even for us normal folks, we have a great influence on later generations. It is amazing to see the mannerisms and hear phrases my grandfather (long passed) used, also used by my grandchildren.
So, I don’t know if there is a supernatural life after death or not. Anyone who is honest must admit the same. But certainly there are things that survive a person, and perhaps that is enough. While I feel Mr. Hawking made an unfortunate analogy, his contributions to humanity give him a greater soul than most.