From $5.5$ months of dual-band optical photometric monitoring at the $1$ mmag level, BRITE-Constellation has revealed two simultaneous types of variability in the O4I(n)fp star $\zeta$ Puppis: one single periodic non-sinusoidal component superimposed on a stochastic component. The monoperiodic component is the $1.78$ d signal previously detected by Coriolis/SMEI, but this time along with a prominent first harmonic. The shape of this signal changes over time, a behaviour that is incompatible with stellar oscillations but consistent with rotational modulation arising from evolving bright surface inhomogeneities. By means of a constrained non-linear light curve inversion algorithm we mapped the locations of the bright surface spots and traced their evolution. Our simultaneous ground-based multi-site spectroscopic monitoring of the star unveiled cyclical modulation of its He II $\lambda4686$ wind emission line with the $1.78$-day rotation period, showing signatures of Corotating Interaction Regions (CIRs) that turn out to be driven by the bright photospheric spots observed by BRITE. Traces of wind clumps are also observed in the He II $\lambda4686$ line and are correlated with the amplitudes of the stochastic component of the light variations probed by BRITE at the photosphere, suggesting that the BRITE observations additionally unveiled the photospheric drivers of wind clumps in $\zeta$ Pup and that the clumping phenomenon starts at the very base of the wind. The origins of both the bright surface inhomogeneities and the stochastic light variations remain unknown, but a subsurface convective zone might play an important role in the generation of these two types of photospheric variability.