It is the immune response that usely causes symptoms with a virus infection. Sickness behavior is part of that.
Yeah they sort of dovetail together. The immune system may be responsible for local symptoms, like for example a congested nose during a cold, chest congestion during a cough, or vomiting and diarrhea during a gastrointestinal infection.
However, when it comes to the general way that you feel when you are sick, with symptoms like fatigue, lethargy, brain fog, low mood, increased pain, these are part of the sickness behavior response, not the immune response. This sickness behavior response is triggered by cytokines from an active immune system, but is distinct from the immune response.
In fact, it is possible to isolate the sickness behavior response from the immune response. The vagus nerve is key to the sickness behavior response: the vagus detects when the immune system is active in the body (which usually indicates an infection is going on). On detection, the vagus nerve then signals to the brain to activate sickness behavior. So this is how the immune response triggers the sickness behavior response.
However, in animals experiments, when the animal's vagus nerve was cut, animals that were abdominally injected with an inflammatory cytokine (IL-1β) normally produced in infections showed all the normal
immune responses to fight the infection, but in spite of the fact that they were injected with this inflammatory cytokine, did not display any of the normal malaise and sickness behavior responses that you would expect during an infectious illness. The animals felt fine, yet their body was mounting an inflammatory response.
So this shows that if it were not for the built-in sickness behavior response, animals and humans would feel perfectly good and healthy, even when they had a nasty case of the flu. They would observe all the local flu symptoms like a runny nose, but they would not feel bad at all. They would feel as fit and able as they usually are.