We present a theoretical study of the band structure and Landau levels in bilayer graphene at low energies in the presence of a transverse magnetic field and Rashba spin-orbit interaction in the regime of negligible trigonal distortion. Within an effective low-energy approach the (Lowdin partitioning theory), we derive an effective Hamiltonian for bilayer graphene that incorporates the influence of the Zeeman effect, the Rashba spin-orbit interaction and, inclusively, the role of the intrinsic spin-orbit interaction on the same footing. Particular attention is paid to the energy spectrum and Landau levels. Our modeling unveils the strong influence of the Rashba coupling lambda(R) in the spin splitting of the electron and hole bands. Graphene bilayers with weak Rashba spin-orbit interaction show a spin splitting linear in momentum and proportional to lambda(R), but scaling inversely proportional to the interlayer hopping energy gamma(1). However, at robust spin-orbit coupling lambda(R), the energy spectrum shows a strong warping behavior near the Dirac points. We find that the bias-induced gap in bilayer graphene decreases with increasing Rashba coupling, a behavior resembling a topological insulator transition. We further predict an unexpected asymmetric spin splitting and crossings of the Landau levels due to the interplay between the Rashba interaction and the external bias voltage. Our results are of relevance for interpreting magnetotransport and infrared cyclotron resonance measurements, including situations of comparatively weak spin-orbit coupling.