We often say "flu" when referring to unrelated respiratory or stomach viruses. Influenza is the true flu, and while there are many shared symptoms, it is much less common and far more debilitating than even the worst of your regular ol' colds or stomach bugs. Symptoms may include high fevers (around 103-104 deg), significant fatigue, body aches, sore throat, cough, congestion, and vomiting and diarrhea in some pediatric cases.
The influenza vaccine cannot cause influenza -- the injectable version in particular either contains a killed ("inactivated") version of the virus, or does not contain the virus at all (referred to as "recombinant"). You can develop a fever and mild aches after getting the vaccine, but these are simply side effects and should resolve within a few days. These side effects are not due to the virus, but your body's inflammatory response to the injection of foreign antibodies. Any symptoms of viral illness such as cough or sneezing is coincidental.
Not only does getting the flu shot provide you with protection from the most common circulating strains of the virus, but it helps to protect others, such as pregnant women, young babies, the elderly, and the immunocompromised, who may be unable to receive the vaccine and who also stand to suffer the worst complications from the virus.