It is said that immunity from a flu vaccine appears after about two weeks. However, from experience, flu usually lasts only a few days. If sufficient antibodies appear only after two weeks shouldn't also the flu last that long? Or is there a difference between how the body reacts to the actual flu virus as opposed to the dead one in the flu jab (like maybe there being more viral particles in an actual infection or something)?

Sorry if naive question.

It is said that immunity from a flu vaccine appears after about two weeks. However, from experience, the flu usually lasts only a few days. 

If sufficient antibodies appear only after two weeks shouldn't the flu also last that long? Or is there a difference between how the body reacts to the actual flu virus as opposed to the dead one in the flu vaccine (like maybe there being more viral particles in an actual infection or something)?