  We compare two versions of the nonlinear $q$-voter model: the original one,
with annealed randomness, and the modified one, with quenched randomness. In
the original model, each voter changes its opinion with a certain probability
$\epsilon$ if the group of influence is not unanimous. In contrast, the
modified version introduces two types of voters that act in a deterministic way
in case of disagreement in the influence group: the fraction $\epsilon$ of
voters always change their current opinion, whereas the rest of them always
maintain it. Although both concepts of randomness lead to the same average
number of opinion changes in the system on the microscopic level, they cause
qualitatively distinct results on the macroscopic level. We focus on the
mean-field description of these models. Our approach relies on the stability
analysis by the linearization technique developed within dynamical system
theory. This approach allows us to derive complete, exact phase diagrams for
both models. The results obtained in this paper indicate that quenched
randomness promotes continuous phase transitions to a greater extent, whereas
annealed randomness favors discontinuous ones. The quenched model also creates
combinations of continuous and discontinuous phase transitions unobserved in
the annealed model, in which the up-down symmetry may be spontaneously broken
inside or outside the hysteresis loop. The analytical results are confirmed by
Monte Carlo simulations carried out on a complete graph.
