Rotationally constrained convection possesses velocity correlations that transport momentum and drive mean flows such as differential rotation. The nature of this transport can be very complex in turbulent flow regimes, where large-scale, coherent vorticity structures and mean flows can be established by smaller scale turbulence through inverse cascades. The dynamics of the highly turbulent solar convection zone therefore may be quite different than in early global-scale numerical models, which were limited by computational resources to nearly laminar flows. Recent progress in high-performance computing technology and ongoing helioseismic investigations of the dynamics of the solar interior have motivated us to develop more sophisticated numerical models of global-scale solar convection. Here we report three-dimensional simulations of compressible, penetrative convection in rotating spherical shells in both laminar and turbulent parameter regimes. The convective structure in the laminar case is dominated by "banana cells," but the turbulent case is much more complex, with an intricate, rapidly evolving downflow network in the upper convection zone and an intermittent, plume-dominated structure in the lower convection zone and overshoot region. Convective patterns generally propagate prograde at low latitudes and retrograde at high latitudes relative to the local rotation. The differential rotation profiles show some similarity with helioseismic determinations of the solar rotation but still exhibit significantly more cylindrical alignment. Strong, intermittent, vortical downflow lanes and plumes play an important dynamical role in turbulent flow regimes and are responsible for significant differences relative to laminar flows with regard to momentum and energy transport and to the structure of the overshoot region at the base of the convection zone.