Conformations of a bottle-brush polymer with two types (A,B) of grafted side chains are studied by molecular dynamics simulations, using a coarse-grained bead spring model with side chains of up to N = 50 effective monomers. Varying the solvent quality and the grafting density, the crossover from the "pearl-necklace" structure to dense cylinders is studied. Whereas for small grafting density, A- and B-chains form separate collapsed chains, at intermediate grafting density, larger "pearls" containing several chains are observed, exhibiting microphase separation between A and B in "dumbbell"-type configurations. At still lamer grafting density, short-range order of "Janus dumbbell"-type is observed. It is argued that because of the quasi-ID character of bottle-brush polymers with stiff backbones, all phase changes occur gradually, and no sharp-phase transitions like in bulk polymer mixtures or block copolymer melts can be observed.