This is a great question.

Spyware is software that sneakily gathers user information through the user's Internet connection without its knowledge like emails, passwords and credit card information. Spyware does not copy itself. Antispyware software will discover and remove spyware from the computer.

A computer virus is a program that is maliciously downloaded (usually through an URL or link attachment) onto your computer without your knowledge and can replicate/copy itself that can crash your entire system. Antivirus programs will scan and remove any potential viruses.