A virulent virus causes symptomatic or severe disease, while temperate, or non-virulent, viruses rarely cause any disease symptoms at all. Influenza viruses are virulent, causing quick and severe flu symptoms, while temperate viruses like cytomegalovirus cause no symptoms at all and often go unnoticed.
Virulent viruses contain genes and proteins that allow them to quickly hijack and infect cells. Once inside, they replicate and kill the host cell upon exit, releasing thousands of progeny in the process. This massive amount of cell death during infection is what causes many symptoms of virulent virus infection. These viruses also cause the immune system to release cytokines to battle the infection, which causes many other symptoms associated with virulence such as fever and swelling.
Temperate viruses infect cells, but often live in the cells for long periods of time and exit without killing the host cell. Humans, animals, and plants are all infected with some temperate viruses. Since these viruses typically don't kill the host cell upon exit, they are able to easily avoid the immune system. Without cell death or an elicited immune response, there usually aren't symptoms associated with these infections. Rarely, however, these viruses can mutate the host cell. For example, human papillomavirus (HPV) does not typically cause infection, but on rare occasion can manifest as genital warts or cervical cancer in women.