Two main modes of star formation are know to control the growth of galaxies: a relatively steady one in disk-like galaxies, defining a tight star formation rate (SFR)-stellar mass sequence, and a starburst mode in outliers to such a sequence which is generally interpreted as driven by merging. Such starburst galaxies are rare but have much higher SFRs, and it is of interest to establish the relative importance of these two modes. PACS/Herschel observations over the whole COSMOS and GOODS-South fields, in conjunction with previous optical/near-IR data, have allowed us to accurately quantify for the first time the relative contribution of the two modes to the global SFR density in the redshift interval 1.5 < z < 2.5, i.e., at the cosmic peak of the star formation activity. The logarithmic distributions of galaxy SFRs at fixed stellar mass are well described by Gaussians, with starburst galaxies representing only a relatively minor deviation that becomes apparent for SFRs more than four times higher than on the main sequence. Such starburst galaxies represent only 2% of mass-selected star-forming galaxies and account for only 10% of the cosmic SFR density at z 2. Only when limited to SFR > 1000 M yr-1, off-sequence sources significantly contribute to the SFR density (46% 20%). We conclude that merger-driven starbursts play a relatively minor role in the formation of stars in galaxies, whereas they may represent a critical phase toward the quenching of star formation and morphological transformation in galaxies.