I understand the regexp. $1 = book.pdf

So for the specific request in the example, we could write:

location ~ ^/download/(.*)$ {
alias /home/website/files/book.pdf;
}

Is it that the example is unrealistic? Because this config does exactly the same thing:
location /download/ {
alias /home/website/files/;
}

I'm struggling to see the difference, and why one would use the regexp.

How about requesting "/download/baz/foo/book.pdf" for the above regexp config and this one:
location ~ ^/download/(.*)/foo/ {
alias /home/website/files/$1;
}

(why would I do this?)