In short: You cannot reliably differentiate between Hantavirus pulmonary syndrome and flu from symptoms alone.
Facts About Hantaviruses (CDC.gov):
Usually, people do not have a runny nose, sore throat, or a rash.
Hantavirus Pulmonary Syndrome in Pregnancy (Clinical Infectious Diseases), case review:
She also noted watery eyes, a runny nose, and a sore throat.
Initial flu symptoms usually have a sudden onset and include high fever and headache (but: fever can be not so high and headache can be mild). Other or later symptoms include muscle pain, dry cough, severe fatigue or nausea.
Some considerations: You get flu from another person, not from rats; incubation period (time from exposure to infection) is few days; in Northern hemisphere, the season is roughly October to March.
Source: Difference between cold and flu (Ehealthstar.com)
Initial symptoms of Hantavirus pulmonary syndrome are fatigue, fever and muscle aches. Other early symptoms can include headache, nausea, diarrhea and abdominal pain. Late symptoms can include cough and shortness of breath.
Source: Hantavirus pulmonary syndrome (CDC.gov)
Consideration: Hantavirus spreads year round but mainly in spring and summer, by inhaling aerosols from rat droppings in farms; incubation period is considered to be 1-5 weeks...
Source: Hantavirus Pulmonary Syndrome --- United States: Updated Recommendations for Risk Reduction (CDC.gov)
So, symptoms are almost the same. Influenza without complications does not affect lungs to such extent that would cause shortness of breath. In individuals with weak immunity, influenza can turn into viral pneumonia, which can cause shortness of breath, though. On the other hand, shortness of breath is one of the late symptoms of Hantavirus infection.