Type II supernovae (SNe II) show strong hydrogen features in their spectra throughout their whole evolution, while type IIb supernovae (SNe IIb) spectra evolve from dominant hydrogen lines at early times to increasingly strong helium features later on. However, it is currently unclear whether the progenitors of these SN types form a continuum in pre-SN hydrogen mass or whether they are physically distinct. SN light-curve morphology directly relates to progenitor and explosion properties such as the amount of hydrogen in the envelope, the pre-SN radius, the explosion energy, and the synthesized mass of radioactive material. In this work, we study the morphology of the optical-wavelength light curves of hydrogen-rich SNe II and hydrogen-poor SNe IIb to test whether an observational continuum exists between the two. Using a sample of 95 SNe (73 SNe II and 22 SNe IIb), we define a range of key observational parameters and present a comparative analysis between both types. We find a lack of events that bridge the observed properties of SNe II and IIb. Light-curve parameters such as rise times and post-maximum decline rates and curvatures clearly separate both SN types and we therefore conclude that there is no continuum, with the two SN types forming two observationally distinct families. In the V band a rise time of 17d (SNe II lower and SNe IIb higher), and a magnitude difference between 30 and 40d post-explosion of 0.4mag (SNe II lower and SNe IIb higher) serve as approximate thresholds to differentiate both types.