We demonstrate gate-tunable resonant tunneling and negative differential resistance between two rotationally aligned bilayer graphene sheets separated by bilayer WSe2. We observe large interlayer current densities of 2 uA/um2 and 2.5 uA/um2, and peak-to-valley ratios approaching 4 and 6 at room temperature and 1.5 K, respectively, values that are comparable to epitaxially grown resonant tunneling heterostructures. An excellent agreement between theoretical calculations using a Lorentzian spectral function for the two-dimensional (2D) quasiparticle states, and the experimental data indicates that the interlayer current stems primarily from energy and in-plane momentum conserving 2D-2D tunneling, with minimal contributions from inelastic or non-momentum-conserving tunneling. We demonstrate narrow tunneling resonances with intrinsic half-widths of 4 and 6 meV at 1.5 K and 300 K, respectively.