Viruses and Public Health
A primary reason today to care about malicious code, such as viruses and worms, is the same reason it is important to keep one's vaccination records up to date while traveling. You would not want to become a carrier of some awful disease. Most malicious code today is concerned not only with trashing your machine, but also in using your machine to infect others.
A classic example is the software used to create a DDoS attack. After hiding itself in your computer, modern malware typically seeks information from you to use to infect others, and it usually finds it in your address book or by prowling your local area network. The malware then stalks its new victims, often by sending an email in your name and infects them as well. This is akin to the way a virus may propagate itself in a living organism, and it may be a separate operation from executing the payloadperforming the dirty work of the viruswhich follows.
Upon command, your machine and all of those you have inadvertently helped infect may then zero in on some target (such as the White House, or a critical server that helps make the Internet function) with tens of thousands of commands and requests coming from all directions. This brings the target to its knees, accomplishing the attacker's goal. To avoid this, every machine must be a healthy one, so that it does not unwittingly infect its neighbor, which may be you.