SARS-CoV-2 is a novel coronavirus, not encountered before by humans. The wide spectrum of clinical expression of SARS-CoV-2 illness suggests that individual immune responses to SARS-CoV-2 play a crucial role in determining the clinical course after first infection. Immunological studies have focused on patients with moderate to severe disease, demonstrating excessive inflammation in tissues and organ damage. In order to understand the basis of the protective immune response in COVID-19, we performed a longitudinal follow-up, flow-cytometric and serological analysis of innate and adaptive immunity in 64 adults with a spectrum of clinical presentations: 28 healthy SARS-CoV-2-negative contacts of COVID-19 cases; 20 asymptomatic SARS-CoV-2-infected cases; eight patients with Mild COVID-19 disease and eight cases of Severe COVID-19 disease. Our data show that high frequency of NK cells and early and transient increase of specific IgA, IgM and, to a lower extent, IgG are associated with asymptomatic SARS-CoV-2 infection. By contrast, monocyte expansion and high and persistent levels of IgA and IgG, produced relatively late in the course of the infection, characterize severe disease. Modest increase of monocytes and different kinetics of antibodies are detected in mild COVID-19. The importance of innate NK cells and the short-lived antibody response of asymptomatic individuals and patients with mild disease suggest that only severe COVID-19 may result in protective memory established by the adaptive immune response.